Geometric $\bm{C^{1}}$ -stability of geodesic spheres under volume-preserving mean curvature flow

Esther Cabezas-Rivas Address: Esther Cabezas-Rivas Universitat de València Departament de Matemàtiques Av. Vicent Andrés Estellés 19 46100 Burjassot Spain esther.cabezas-rivas@uv.es and Carlo Sinestrari Address: Carlo Sinestrari Università degli Studi di Roma ”Tor Vergata” Dipartimento di Matematica Via della Ricerca Scientifica 00133, Roma Italy sinestra@mat.uniroma2.it

Date: October 14, 2026. This work has been partially supported by project PID2025-174652NB-I00, funded by MICIU/AEI /10.13039/501100011033 and by ERDF, EU. The first author has also been partially supported by project CIAICO/2023/035, funded by the Conselleria d’Educació, Cultura, Universitats i Ocupació. The second author has been partially supported by MUR (Ministero dell’Università e della Ricerca) Project MatMod@TOV 2037-2027, CUP E83C23000330006 and by the MUR Prin 2022 Project CUP E53D23005750006. The second author is a member of INdAM (Istituto Nazionale di Alta Matematica).

Abstract.

We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed.

The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre.

For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.

Key words and phrases:

Mean curvature flow; stability; volume-preserving; Alexandrov reflection

2020 Mathematics Subject Classification

53E10, 35B35, 53C21, 35B40

1. Introduction and main results

Volume-preserving mean curvature flow is the constrained gradient flow of area under fixed enclosed volume. Its equilibria are closed constant-mean-curvature hypersurfaces, including round spheres in Euclidean space. A fundamental problem is to identify conditions on the initial data that prevent singularity formation and force the flow to exist for all time and converge to a steady state. Classical convergence results [25, 18, 27, 3, 9] typically require convexity, pointwise or integral curvature pinching, or closeness to a sphere in a topology that quantitatively controls the variation of the normal.

In this paper, we show that the stability of geodesic spheres can be established under purely first-order geometric assumptions. The initial hypersurface is required only to lie in a sufficiently thin annulus around a geodesic sphere and to have outward unit normal making a small angle with the radial direction. No convexity or mean convexity is assumed, and no initial curvature bound or quantitative modulus of continuity of the normal is imposed. We prove that the corresponding volume-preserving mean curvature flow exists for all time, becomes strictly convex, and converges smoothly and exponentially to a geodesic sphere. We also extend the construction to Lipschitz initial hypersurfaces by deriving positive-time smoothing estimates that are uniform along smooth approximations.

To state our results precisely, let $M$ be a smooth, closed, $n$ -dimensional manifold and let $X_{0}:M\rightarrow\mathbb{R}^{n+1}$ be a smooth embedding. We denote by $M_{0}=X_{0}(M)$ the corresponding embedded hypersurface and assume that $M_{0}$ bounds a domain $\Omega_{0}$ . Let $R_{0}>0$ be determined by $\operatorname{vol}(\Omega_{0})=\omega_{n+1}R_{0}^{n+1},$ where $\omega_{n+1}$ denotes the volume of the unit ball in $\mathbb{R}^{n+1}$ .

We assume that $M_{0}$ is $C^{1}$ -close to the sphere $\partial B_{R_{0}}(0)$ in the following sense: there exist $\varepsilon_{0}\in(0,1)$ and $\theta_{0}\in(0,\pi/2)$ such that

(1.1)

R_{0}(1-\varepsilon_{0})\leq|x|\leq R_{0}(1+\varepsilon_{0})\qquad\text{for every }x\in M_{0},

and

(1.2)

\left\langle\nu_{0}(x),\frac{x}{|x|}\right\rangle\geq\cos\theta_{0}\qquad\text{for every }x\in M_{0},

where $\nu_{0}$ is the outward unit normal to $M_{0}$ .

We refer to (1.1) and (1.2) as the annular-closeness and the normal-oscillation assumption, respectively. They can be regarded as a geometric formulation of $C^{1}$ -closeness to $\partial B_{R_{0}}(0)$ . Indeed, (1.2) implies that $M_{0}$ is star-shaped with respect to the origin, so that $M_{0}$ is a radial graph over $\mathbb{S}^{n}$ ; then (1.1) and (1.2) are equivalent to a bound on the height and gradient of its graph function.

We consider the volume-preserving mean curvature flow

(1.3)

\partial_{t}X(p,t)=\bigl(h(t)-H(p,t)\bigr)\nu(p,t),\qquad p\in M,\quad t>0,

with $X(\cdot,0)=X_{0}$ , where $h(t)$ is the averaged mean curvature, that is,

(1.4)

h(t)=\frac{1}{|M_{t}|}\int_{M_{t}}H\,d\mu_{t}.

Here $M_{t}=X(M,t)$ , while $H$ and $\nu$ denote its mean curvature and outward unit normal, respectively. Our sign convention is such that $-H\nu$ points inwards on a convex hypersurface (i.e. with $\kappa_{1}\geq 0$ , where $\kappa_{1}\leq\cdots\leq\kappa_{n}$ denote the principal curvatures).

Our stability result in the Euclidean setting reads as follows.

Theorem 1.1.

For every $n\geq 1$ , there exists $\delta_{*}=\delta_{*}(n)>0$ with the following property. Let $M_{0}\subset\mathbb{R}^{n+1}$ satisfy (1.1) and (1.2) with $\varepsilon_{0}+\theta_{0}<\delta_{*}.$ Then (1.3) satisfies the following:

(a)

It has a unique smooth solution, defined for every $t\geq 0$ .
(b)

The hypersurfaces $M_{t}$ remain uniformly star-shaped and satisfy uniform curvature bounds.
(c)

There exists $T<\infty$ such that $M_{t}$ is strictly convex for every $t\geq T$ .

(d)

As $t\to\infty$ , $M_{t}$ converges exponentially fast in the $C^{\infty}$ topology to a round sphere $B_{R_{0}}(p_{\infty})$ for some $p_{\infty}\in\mathbb{R}^{n+1}$ . More precisely, for every $k\in\mathbb{N}$ , there exist constants $C_{k},\lambda_{k}>0$ such that

\operatorname{dist}_{C^{k}}\bigl(M_{t},\partial B_{R_{0}}(p_{\infty})\bigr)\leq C_{k}e^{-\lambda_{k}t}\qquad\text{ for all sufficiently large $t$. }

(e)

The centre of the limiting sphere satisfies

|p_{\infty}|\leq\frac{R_{0}}{2}\Phi(\varepsilon_{0},\theta_{0}),\qquad\text{where}\quad\Phi(\varepsilon,\theta)=(1+\varepsilon)\sin\theta+\sqrt{(1+\varepsilon)^{2}\sin^{2}\theta+4\varepsilon}.

In particular, $|p_{\infty}|\leq R_{0}\bigl(2\theta_{0}+\sqrt{\varepsilon_{0}}\bigr)$ for sufficiently small initial parameters, giving a control on the displacement from the reference centre.

Our method of proof is based on the Alexandrov reflection technique, adapted to the setting of geometric flows given by Chow and Gulliver [13]. The approach is therefore direct and geometric compared with some of the techniques of the previous literature, e.g. integral estimates [25, 27] or the functional analytic techniques employed in [18].

The Euclidean result contains the core reflection mechanism, but extending it to the nonflat space forms requires additional geometric input. Alexandrov reflection remains available after replacing affine hyperplanes by totally geodesic hypersurfaces. In the spherical case, however, the reflection construction is necessarily restricted to a hemisphere, while in both nonflat settings the evolution equations contain ambient-curvature terms that must be incorporated into the curvature estimates and the late-time analysis.

Hence, with some additional work, we can extend our $C^{1}$ -stability result to the non-flat setting as follows. Let $N_{K}^{n+1}$ denote the simply connected space form of constant sectional curvature $K\in\{-1,1\}$ , and fix $o\in N_{K}^{n+1}$ . Now $B_{R}^{K}(p)$ represents the geodesic ball of radius $R$ centred at $p\in N_{K}^{n+1}$ . Consider $M_{0}\subset N_{K}^{n+1}$ a smooth, closed and embedded hypersurface enclosing a domain $\Omega_{0}$ , and let $R_{0}>0$ be determined by $\operatorname{vol}(\Omega_{0})=\operatorname{vol}\bigl(B_{R_{0}}^{K}(o)\bigr).$ Writing $r(x)=\operatorname{dist}_{N_{K}}(o,x),$ we assume that

(1.5)

R_{0}(1-\varepsilon_{0})\leq r(x)\leq R_{0}(1+\varepsilon_{0})\qquad\text{for every }x\in M_{0},

(1.6)

\left\langle\nu_{0},\partial_{r}\right\rangle\geq\cos\theta_{0}\qquad\text{on }M_{0}.

Theorem 1.2 (Nonflat space-form extension).

Let $1\leq n\in\mathbb{N}$ , $K\in\{-1,1\}$ , and let $M_{0}\subset N_{K}^{n+1}$ satisfy (1.5) and (1.6). In the spherical case, assume also that $R_{0}(1+\varepsilon_{0})<\frac{\pi}{2}.$ If $\varepsilon_{0}$ and $\theta_{0}$ are sufficiently small, then conclusions (a)–(d) of Theorem 1.1 hold, with round spheres replaced by geodesic spheres in $N_{K}^{n+1}$ . In particular, the limiting hypersurface is $\partial B_{R_{0}}^{K}(p_{\infty})$ for some $p_{\infty}\in N_{K}^{n+1}$ . In addition, when $K=-1$ , the evolving hypersurfaces become strictly horospherically convex for all sufficiently large times. The conclusion about the centre is now

\operatorname{dist}_{N_{K}}(o,p_{\infty})\leq\omega_{K}(R_{0},\varepsilon_{0},\theta_{0}),

for a suitable positive function $\omega_{K}$ satisfying $\omega_{K}(R_{0},\varepsilon_{0},\theta_{0})\to 0$ as $\varepsilon_{0}+\theta_{0}\to 0$ .

The use of the initial assumptions in the reflection technique requires a different argument in the non-flat case, and the estimates are not as direct and explicit as in Euclidean space. This leads to the different formulation of the displacement estimate at the end of the statement.

The geometric assumptions in the above results involve only the radial position and the radial tilt of the initial hypersurface, and therefore remain meaningful for Lipschitz radial graphs. By combining smooth approximation with uniform curvature estimates, we extend the stability result to this rough-data setting as follows.

Theorem 1.3 (Lipschitz initial hypersurfaces).

Let $1\leq n\in\mathbb{N}$ , $K\in\{-1,0,1\}$ , and $M_{0}\subset N_{K}^{n+1}$ be a closed embedded Lipschitz hypersurface that satisfies the geometric assumptions of Theorem 1.1 or Theorem 1.2, with the normal-oscillation condition understood almost everywhere. Then there exists an instantaneously smooth volume-preserving mean curvature flow $(M_{t})_{t>0},$ with initial condition $M_{0}$ in the following continuous sense: writing $M_{t}$ and $M_{0}$ as radial graphs with respect to the reference point $o$ , with radial functions $\rho(\cdot,t)$ and $\rho_{0}$ , respectively, one has

\rho(\cdot,t)\longrightarrow\rho_{0}\qquad\text{uniformly as }t\searrow 0.

In particular, $M_{t}$ converges to $M_{0}$ in the Hausdorff sense and encloses the same volume as $M_{0}$ . The flow has the same eventual convexity and exponential convergence properties as in the corresponding smooth theorems.

Relation to previous work and main novelty

There is a rich literature on convergence for the flow (1.3). Huisken [25] proved global existence and convergence to a round sphere for convex hypersurfaces in $\mathbb{R}^{n+1}$ , $n\geq 2$ , while the corresponding result for curves is due to Gage [20]. A different line of research, initiated by Escher and Simonett [18], replaces convexity by perturbative closeness to a sphere in the little Hölder space $h^{1+\alpha}$ , using centre-manifold and maximal-regularity methods. Similar stability results were obtained under Sobolev smallness assumptions in [7], under integral curvature pinching in [27], and, for strictly stable critical sets in flat tori, under $C^{1,1}$ -small perturbations in [16].

Related results in nonflat space forms rely on stronger geometric or functional hypotheses. In hyperbolic space, the first author and Miquel [9] proved convergence from horospherically convex hypersurfaces and from $C^{1,\alpha}$ -perturbations of geodesic spheres, while subsequent works [23, 35, 28] considered $L^{2}$ -almost umbilical initial hypersurfaces. Hartley [21] treated constrained curvature flows near geodesic spheres in $\mathbb{S}^{n+1}$ , under smallness assumptions in spaces $h^{2+\alpha}$ . For curves on pinched Hadamard surfaces (i.e. allowing some variable negative ambient curvature), [1, Theorem 1.3] established stability of geodesic circles under $h^{1+\beta}$ -small perturbations.

All the aforementioned results either impose bounds on the curvature, in a pointwise or integral sense, or consider perturbations of a sphere in a functional norm which implies at least $C^{1,\alpha}$ smallness. Our hypotheses instead control only the radial position of the hypersurface and the pointwise angle of its normal with the radial direction. In particular, they impose no direct bound on the second fundamental form and no Hölder control of the normal, whose spatial variation may be arbitrarily rapid. The first-order nature of our assumptions is essential. Mere $C^{0}$ -closeness does not prevent singularities: one may attach to a sphere an arbitrarily small bubble through a sufficiently thin neck, producing a hypersurface arbitrarily close to the sphere in $C^{0}$ whose flow develops a neckpinch.

Our approach is based on quantitative Alexandrov reflection, in the form developed for geometric flows by Chow and Gulliver [13, 12]. Reflection has been used mainly for expanding flows and, more recently, for constrained flows in Euclidean and hyperbolic space, in combination with convexity or related positive-curvature assumptions; see [8, 22, 6, 4, 5, 34]. Here, by contrast, quantitative reflection and geometric $C^{1}$ -closeness determine each other, up to a change of parameters. The preserved reflection threshold therefore controls both the first-order geometry and the displacement of the limiting centre from the prescribed reference centre.

For more general ambient manifolds, convergence to constant-mean-curvature hypersurfaces has been established for suitably $C^{2}$ -round or non-degenerate perturbations of small geodesic spheres [26, 31, 3, 30]. In this generality, reflection symmetries may be unavailable and closed constant-mean-curvature hypersurfaces may be unstable. The stability studied here is therefore tied to the geometry of the ambient space.

Strategy of the proof

The fundamental step is to establish a quantitative relation between our geometric assumptions and a strong Alexandrov reflection property. Proposition 3.1 shows that the initial annular-closeness and normal-oscillation assumptions imply reflection across hyperplanes in every direction up to a common threshold, located at a small distance from the center of the reference sphere. Conversely, Corollary 3.5 shows that, within the volume-normalized class, such a threshold controls both the radial oscillation and the tilt of the hypersurface. Thus, up to a quantitative change of parameters, strong reflection and geometric $C^{1}$ -closeness determine each other.

The decisive advantage of the reflection formulation is its invariance under the flow. By the reflection principle of Chow and Gulliver [13], the initial reflection property is preserved for all positive times. Consequently, the evolving hypersurface remains in a fixed annular region and its normal remains uniformly close to the radial direction. These preserved first-order bounds provide the geometric input needed to recover the missing curvature estimates.

More precisely, the radial and angular estimates give a two-sided bound for the support function, allowing us to adapt Tso’s maximum-principle argument [33] to control the mean curvature. This step is more delicate than in the standard convex setting, since neither $H$ nor its spatial average $h$ has a prescribed sign. The full second fundamental form is then controlled in Proposition 4.4 by applying the maximum principle to a weighted quantity as in Ecker-Huisken’s interior estimates [17]. Since the tilt is measured relative to the spatially varying radial direction rather than to a fixed ambient vector, its evolution produces additional radial terms. The radial and angular bounds provided by Alexandrov reflection allow us to control these contributions.

Global existence is then obtained by parabolic regularity and a continuation argument. The area dissipation implies smooth subsequential convergence to a closed constant-mean-curvature hypersurface, which is a sphere by Alexandrov’s theorem; volume preservation determines its radius. A sufficiently late time slice is therefore uniformly convex, allowing the flow to enter the classical stability regime and giving exponential convergence to a sphere. Finally, the preserved reflection property passes to the limiting ball and yields the quantitative estimate for its centre.

The same overall strategy applies to nonflat space forms, with the Euclidean position vector replaced by the radial conformal vector field $s_{K}(r)\partial_{r}.$ The initial reflection threshold, however, must be estimated by a different argument. We first analyse it quantitatively for an off-centred geodesic sphere and then extend the estimate to hypersurfaces satisfying the annular and normal-oscillation assumptions by a perturbative continuity argument. This yields the radial and normal estimates needed for the curvature argument. The evolution equations now contain additional ambient-curvature terms, whose control uses the same preserved first-order geometry but requires separate estimates.

Although the reflection trick is common to the three space forms, the late-time analysis is not. In hyperbolic space, smooth subsequential convergence places the flow in the strictly horospherically convex regime, and the conclusion follows from the convergence theorem of [9]. In the spherical case, convexity alone does not provide an analogous route to convergence. We therefore develop a separate self-contained bootstrap. For $n\geq 2$ , it is based on the evolution of the curvature pinching and the favourable sign of the relevant ambient-curvature contribution near a geodesic sphere; for curves, the corresponding conclusion is obtained by an energy argument.

The passage to Lipschitz initial data requires a separate curvature argument. Smooth volume-normalized approximations retain the same uniform first-order geometric bounds, but the mean-curvature estimate used in the smooth theory is not uniform because its constant depends on the initial curvature. We therefore bypass that estimate and apply a modified weighted maximum-principle argument directly to the full second fundamental form. The resulting positive-time curvature estimates are uniform along the approximating sequence and yield smooth compactness away from the initial time. The integrability of the radial velocity near $t=0$ then shows that the limit flow converges to the prescribed initial hypersurface and encloses the same volume. The argument of the smooth case then gives the same exponential convergence to a sphere.

Organization of the paper

Section 2 collects the evolution equations and introduces the Alexandrov reflection framework. Section 3 establishes the quantitative correspondence between the Euclidean reflection property and the radial and normal estimates, and shows that the latter are preserved along the flow. Section 4 derives the bounds for the mean curvature and the full second fundamental form. Section 5 proves long-time existence, eventual convexity, smooth exponential convergence, and the quantitative estimate for the centre of the limiting sphere in Euclidean space. Section 6 extends the argument to hyperbolic and spherical space forms, including the ambient-curvature terms and the separate late-time analysis required in each setting. Finally, Section 7 treats Lipschitz initial hypersurfaces by smooth approximation and establishes the uniform positive-time estimates required for instantaneous regularization and the subsequent long-time behaviour.

2. Notation, conventions and preliminary results

Throughout the paper, $M_{t}$ is a time-dependent closed embedded hypersurface in a space form, $g=(g_{ij})$ denotes the induced metric on $M_{t}$ , $\nabla$ its Levi-Civita connection, and $\Delta$ the Laplace–Beltrami operator. We denote by $A=(h_{i}^{j})$ the Weingarten operator with respect to the outward unit normal $\nu$ , and by $H=\operatorname{tr}_{g}A$ the mean curvature. Our sign convention is such that the principal curvatures of the outward-oriented sphere $\partial B_{R}(p)$ are all positive.

Indices are raised and lowered using $g$ , and repeated indices above and below are summed. We write

|A|^{2}=h_{i}^{j}h_{j}^{i},\qquad\operatorname{tr}(A^{3})=h_{i}^{j}h_{j}^{k}h_{k}^{i}.

The notation $S*T$ will denote a linear combination of contractions of the tensors $S$ and $T$ with respect to $g$ . Constants denoted by $C$ may change from line to line and depend only on the fixed parameters and the initial hypersurface, unless otherwise stated.

In the first part of the paper, we consider the flow in a Euclidean ambient space. We collect the evolution equations which we will later use in the curvature estimates. All geometric quantities below are computed on $M_{t}$ , and the time derivative is taken in the parametrization determined by (1.3).

Lemma 2.1 (Evolution equations).

Under the vp-mcf (1.3) in $\mathbb{R}^{n+1}$ , the following holds:

(a)

$(\partial_{t}-\Delta)H=(H-h)|A|^{2}$ .
(b)

The support function $u=\langle X,\nu\rangle$ satisfies $(\partial_{t}-\Delta)u=h-2H+|A|^{2}u$ .
(c)

$(\partial_{t}-\Delta)|A|^{2}=-2|\nabla A|^{2}+2|A|^{4}-2h\operatorname{tr}(A^{3}).$

Following [13], we introduce the terminology for reflection of a closed hypersurface across a hyperplane.

Definition 2.2 (Reflection across a hyperplane).

Let $M\subset\mathbb{R}^{n+1}$ be a smooth, closed, embedded hypersurface bounding an open domain $\Omega$ , and let

\Pi=\Pi_{\lambda,e}:=\left\{x\in\mathbb{R}^{n+1}:\langle x,e\rangle=\lambda\right\},

for given $e\in\mathbb{S}^{n}$ and $\lambda\in\mathbb{R}$ . We denote the two open half-spaces determined by $\Pi$ by

\Pi^{+}:=\left\{x\in\mathbb{R}^{n+1}:\langle x,e\rangle>\lambda\right\},\qquad\Pi^{-}:=\left\{x\in\mathbb{R}^{n+1}:\langle x,e\rangle<\lambda\right\}.

(a)

The reflection across $\Pi$ is the Euclidean isometry

$\mathcal{R}_{\Pi}(x)=x-2\bigl(\langle x,e\rangle-\lambda\bigr)e.$

In particular, $\mathcal{R}_{\Pi}(\Pi^{+})=\Pi^{-}$ , $\mathcal{R}_{\Pi}(\Pi^{-})=\Pi^{+}$ , and $\mathcal{R}_{\Pi}$ fixes every point of $\Pi$ .
(b)

$M$ reflects across $\Pi$ in the direction $e$ if the reflection of the portion of $\Omega$ lying in $\Pi^{+}$ is contained in $\Omega$ , namely

(2.1) $\mathcal{R}_{\Pi}\bigl(\Omega\cap\Pi^{+}\bigr)\subset\Omega\cap\Pi^{-}.$

Equivalently, the reflected cap $\mathcal{R}_{\Pi}\bigl(M\cap\Pi^{+}\bigr)$ does not cross $M$ and lies in $\overline{\Omega}\cap\Pi^{-}$ .
(c)

$M$ reflects strictly across $\Pi$ if

$\mathcal{R}_{\Pi}\bigl(M\cap\Pi^{+}\bigr)\subset\Omega\cap\Pi^{-}$

and if the direction $e$ is not tangent to $M$ at any point of $M\cap\Pi$ .
(d)

$M$ reflects strictly up to $\Pi_{\lambda,e}$ if it reflects strictly across every parallel hyperplane $\Pi_{\mu,e}$ with $\mu\geq\lambda$ .

In the present paper, the hyperplanes $\Pi_{\lambda,e}$ are always taken with $\lambda>0$ and therefore $\Pi^{+}$ is the side of the reflecting hyperplane which is farther from the origin.

Reflection across a hyperplane is an ambient isometry and therefore preserves the local geometric terms in the evolution equation. Moreover, the original and reflected hypersurfaces carry the same spatially constant term $h(t)$ , which cancels when their local graph equations are compared. Hence the interior and boundary tangency arguments in [13, Theorem 2.2] apply without change, and we obtain the following result.

Theorem 2.3 (Preservation of the reflection property; cf. Chow–Gulliver).

Let $M_{t}\subset\mathbb{R}^{n+1}$ be a solution of the volume-preserving mean curvature flow (1.3), for $t\in[0,T)$ . If $M_{0}$ reflects strictly across (resp. up to) $\Pi_{\lambda,e}$ for some $e\in\mathbb{S}^{n}$ and $\lambda\in\mathbb{R}$ , then $M_{t}$ also reflects strictly across (resp. up to) $\Pi_{\lambda,e}$ , for all $t\in[0,T)$ .

3. $C^{1}$ -control along the flow via Alexandrov reflection

3.1. Alexandrov reflection property at the initial time

We first show that our assumptions on $M_{0}$ imply the reflection property up to a certain threshold, whose size can be estimated in terms of the constants appearing in (1.1) and (1.2).

Proposition 3.1.

Let $c_{0}>0$ satisfy

(3.1)

\frac{2c_{0}}{R_{0}}>\Phi(\varepsilon_{0},\theta_{0}):=(1+\varepsilon_{0})\sin\theta_{0}+\sqrt{(1+\varepsilon_{0})^{2}\sin^{2}\theta_{0}+4\varepsilon_{0}}.\textbf{}

Then $M_{0}$ reflects strictly up to $\Pi=\Pi_{c_{0},e}$ , for every $e\in\mathbb{S}^{n}$ .

Proof.

After a rotation we may assume that

\Pi=\{x\in\mathbb{R}^{n+1}:\langle x,e_{n+1}\rangle=c_{0}\}.

The goal is to show that the part of $M_{0}$ lying above the hyperplane $\Pi$ reflects into the domain enclosed by $M_{0}$ .

It is convenient to write points $x\in\mathbb{R}^{n+1}$ as $x=(\bar{x},x_{n+1})\in\mathbb{R}^{n}\times\mathbb{R}$ . So let us take any $x=(\bar{x},x_{n+1})\in M_{0}$ with $x_{n+1}>c_{0}$ , and denote by

\widetilde{x}=(\bar{x},2c_{0}-x_{n+1})

its reflection across $\Pi$ . Suppose, in order to get a contradiction, that $\widetilde{x}\in M_{0}$ . By the Hausdorff closeness assumption (1.1), we can write

R_{0}^{2}(1-\varepsilon_{0})^{2}\leq|\widetilde{x}|^{2}=|x|^{2}-4c_{0}(x_{n+1}-c_{0})\leq R_{0}^{2}(1+\varepsilon_{0})^{2}-4c_{0}(x_{n+1}-c_{0}).

Hence

4c_{0}(x_{n+1}-c_{0})\leq R_{0}^{2}\bigl((1+\varepsilon_{0})^{2}-(1-\varepsilon_{0})^{2}\bigr)=4\varepsilon_{0}R_{0}^{2},

and therefore

c_{0}\leq x_{n+1}\leq c_{0}+\frac{\varepsilon_{0}R_{0}^{2}}{c_{0}}.

Consequently, $\widetilde{x}_{n+1}=2c_{0}-x_{n+1}$ satisfies

c_{1}:=c_{0}-\frac{\varepsilon_{0}R_{0}^{2}}{c_{0}}\leq\widetilde{x}_{n+1}\leq c_{0}.

Hence, an intersection between $M_{0}$ and its reflection can only occur at points with height at least $c_{1}$ . Observe that our assumption on $c_{0}$ implies $2c_{0}>R_{0}\sqrt{4\varepsilon_{0}}$ , hence $c_{1}>0$ .

We want to show that $M_{0}\cap\{x_{n+1}\geq c_{1}\}$ is a graph over the hyperplane $\{x_{n+1}=0\}$ . Let $z\in M_{0}\cap\{x_{n+1}\geq c_{1}\}$ , and denote by $\xi=z/|z|$ the radial direction. Since assumption (1.2) ensures that

\left\langle\nu(z),\xi\right\rangle\geq\cos\theta_{0},

the angle between $\nu(z)$ and $\xi$ is at most $\theta_{0}$ . Now we can write

\nu(z)=\cos\alpha\,\xi+\sin\alpha\,\eta,

where $\eta$ is a unit vector orthogonal to $\xi$ , and $\alpha\in[0,\theta_{0}]$ denotes the angle between $\nu(z)$ and $\xi$ . Decomposing $e_{n+1}$ into its components parallel and orthogonal to $\xi$ , we have

e_{n+1}=\frac{z_{n+1}}{|z|}\xi+\sqrt{1-\frac{z_{n+1}^{2}}{|z|^{2}}}\,\xi^{\perp},

for some unit vector $\xi^{\perp}\perp\xi$ . Using Cauchy-Schwarz, we deduce

\displaystyle\langle\nu(z),e_{n+1}\rangle\geq\frac{z_{n+1}}{|z|}\cos\alpha-\sqrt{1-\frac{z_{n+1}^{2}}{|z|^{2}}}\sin\alpha.

Since $z_{n+1}>0$ , the right-hand side is decreasing as a function of $\alpha\in[0,\theta_{0}]$ . Hence

(3.2)

\langle\nu(z),e_{n+1}\rangle\geq\frac{z_{n+1}}{|z|}\cos\theta_{0}-\sqrt{1-\frac{z_{n+1}^{2}}{|z|^{2}}}\sin\theta_{0}.

The right-hand side is positive provided

\frac{z_{n+1}}{|z|}>\sin\theta_{0}.

Using $z_{n+1}\geq c_{1}$ and $|z|\leq R_{0}(1+\varepsilon_{0})$ , and recalling the definition of $c_{1}$ , we see from (3.2) that $\langle\nu(z),e_{n+1}\rangle>0$ provided

(3.3)

c_{0}-\dfrac{\varepsilon_{0}R_{0}^{2}}{c_{0}}>{R_{0}(1+\varepsilon_{0})}\sin\theta_{0}.

Setting $y=\frac{c_{0}}{R_{0}}$ , we may rewrite (3.3) as $y^{2}-(1+\varepsilon_{0})\sin\theta_{0}\,y-\varepsilon_{0}>0.$ The positive root of the corresponding quadratic polynomial is

\frac{(1+\varepsilon_{0})\sin\theta_{0}+\sqrt{(1+\varepsilon_{0})^{2}\sin^{2}\theta_{0}+4\varepsilon_{0}}}{2}.

Therefore (3.3) follows from the assumed lower bound on $c_{0}$ . We have proved that

(3.4)

\langle\nu(z),e_{n+1}\rangle>0\qquad\text{on }\qquad M_{0}\cap\{x_{n+1}\geq c_{1}\}.

Now we claim that (3.4) implies that $M_{0}\cap\{x_{n+1}\geq c_{1}\}$ is a graph over the horizontal hyperplane. Indeed, the condition $\langle\nu,e_{n+1}\rangle>0$ shows first that the vertical projection is a local diffeomorphism. To prove global injectivity, let $\Omega_{0}$ be the bounded domain enclosed by $M_{0}$ and suppose that a vertical line

L_{\bar{z}}=\{(\bar{z},s):s\in\mathbb{R}\}

meets $M_{0}\cap\{x_{n+1}\geq c_{1}\}$ in two distinct points

z^{-}=(\bar{z},s_{-}),\qquad z^{+}=(\bar{z},s_{+}),\qquad c_{1}\leq s_{-}<s_{+}.

Both intersections are transverse. Moreover, $\langle\nu,e_{n+1}\rangle>0$ means that, as the line is crossed in the direction of increasing $s$ , both intersections are exit points from $\Omega_{0}$ . This is impossible, since the transverse intersections of the boundary of a bounded domain with an oriented line alternate between entry points, where $\langle\nu,e_{n+1}\rangle<0$ , and exit points, where $\langle\nu,e_{n+1}\rangle>0$ . Any intermediate entry point would also have height at least $c_{1}$ , contradicting the positivity of $\langle\nu,e_{n+1}\rangle$ on the whole portion. Thus every vertical line meets this portion at most once, and the portion is a graph.

Since the vertical segment joining $x$ to $\tilde{x}$ is contained in $\{x_{n+1}\geq c_{1}\}$ , the graph property implies that its interior does not meet $M_{0}$ . Moreover, $\langle\nu(x),e_{n+1}\rangle>0$ shows that the segment enters $\Omega_{0}$ immediately below $x$ . Hence its interior lies in $\Omega_{0}$ , and, as $\tilde{x}\notin M_{0}$ , also $\tilde{x}\in\Omega_{0}$ . Thus

\mathcal{R}_{\Pi}\bigl(M_{0}\cap\Pi^{+}\bigr)\subset\Omega_{0}\cap\Pi^{-}.

Finally, we observe that property (3.4) implies that the normal direction to $\Pi$ is not tangent to $M_{0}$ at the points of $M_{0}\cap\Pi$ . We conclude that $M_{0}$ reflects strictly across $\Pi_{c_{0},e}$ . Since the inequality required on $c_{0}$ is also satisfied by the values greater than $c_{0}$ , the property holds for these values too, and we have the strict reflection of $M_{0}$ up to $\Pi_{c_{0},e}$ as in Definition 2.2 (d). ∎

Remark 3.2.

We observe that $\Phi(\varepsilon_{0},\theta_{0})\to 0$ as $\varepsilon_{0}+\theta_{0}\to 0$ . Therefore we can make the reflection threshold $c_{0}$ as small as we wish compared to $R_{0}$ by imposing that $\varepsilon_{0}+\theta_{0}$ are sufficiently small.

3.2. Reflection and first-order closeness to spheres

Before considering the evolution, we isolate the flow-independent geometric consequences of strong Alexandrov reflection. In the previous subsection, we have shown that geometric $C^{1}$ -closeness to a sphere yields the reflection property across hyperplanes with distance from the origin greater than a small threshold $c_{0}$ . By adapting ideas from [13], we now show that the converse is also true, and that the reflection up to a sufficiently small $c_{0}$ gives a control both on the radial oscillation and on the tilt of the hypersurface. The reflection threshold $c_{0}$ may therefore be regarded as a geometric measure of first-order deviation from a sphere.

Proposition 3.3 (Radial control from reflection).

Let $M=\partial\Omega\subset\mathbb{R}^{n+1}$ be a smooth, closed, connected, embedded hypersurface, with $0\in\Omega$ , and let $R_{0}>0$ be such that $\Omega$ has the same volume as $B_{R_{0}}(0)$ . Suppose that there exists $c_{0}\in(0,R_{0}/2)$ such that $M$ reflects across $\Pi_{c_{0},e}$ for every $e\in\mathbb{S}^{n}$ . Then

(3.5)

R_{0}-2c_{0}\leq|x|\leq R_{0}+2c_{0}\qquad\text{on }M.

Proof.

Set

r_{-}=\min_{x\in M}|x|\qquad\text{and}\qquad r_{+}=\max_{x\in M}|x|,

and choose $x_{-},x_{+}\in M$ so that $|x_{-}|=r_{-}$ , $|x_{+}|=r_{+}$ . Then the reflection property of $M$ across $\Pi_{c_{0},e}$ , with $e$ parallel to $x_{+}-x_{-}$ (see the proof of [13, Corollary 2.9]) yields the inequality

(3.6)

r_{+}^{2}-r_{-}^{2}\leq 2c_{0}\,|x_{+}-x_{-}|.

The triangle inequality gives

\displaystyle|x_{+}-x_{-}|

\displaystyle\leq|x_{+}|+|x_{-}|=r_{+}+r_{-}.

Therefore, (3.6) implies

\displaystyle\bigl(r_{+}-r_{-}\bigr)\bigl(r_{+}+r_{-}\bigr)\leq 2c_{0}\bigl(r_{+}+r_{-}\bigr).

Since $M$ is nonempty, $r_{+}>0$ , and hence $r_{+}+r_{-}>0.$ We may therefore divide by the latter and conclude that

(3.7)

r_{+}-r_{-}\leq 2c_{0}.

It remains to locate this annulus relative to $R_{0}$ . Notice that $0\in\Omega$ gives

B_{r_{-}}(0)\subset\Omega\subset B_{r_{+}}(0).

The volume normalization therefore implies $r_{-}\leq R_{0}\leq r_{+}.$ Combining this and (3.7), we find

(3.8)

R_{0}-2c_{0}\leq r_{+}-2c_{0}\leq r_{-}\leq R_{0}\leq r_{+}\leq r_{-}+2c_{0}\leq R_{0}+2c_{0}.

Since $R_{0}>2c_{0}$ by assumption, the lower bound above is strictly positive. Therefore $M$ cannot pass through the origin. Thus, for every $x\in M$ , $r_{-}\leq|x|\leq r_{+},$ and hence

R_{0}-2c_{0}\leq|x|\leq R_{0}+2c_{0},

which finishes the proof. ∎

Proposition 3.4 (Normal control from reflection).

Let $M$ and $R_{0}$ be as in Proposition 3.3. Suppose that $c_{0}\in(0,R_{0}/3)$ is such that $M$ reflects strictly up to $\Pi_{c_{0},e}$ for every $e\in\mathbb{S}^{n}$ . Then

(3.9)

\left\langle\nu,\frac{x}{|x|}\right\rangle^{2}\geq 1-\left(\frac{c_{0}}{R_{0}-2c_{0}}\right)^{2}\qquad\text{on }M.

In particular, $M$ is star-shaped with respect to the origin.

Proof.

By definition of strict reflection, for any $x\in M$ and $e\in\mathbb{S}^{n}$ we have

(3.10)

e\notin T_{x}M\qquad\text{whenever}\qquad\langle x,e\rangle>c_{0}.

For any $x\in M$ , we decompose the position vector into its normal and tangential components:

x=\langle x,\nu\rangle\nu+x^{\top},\qquad\text{with}\quad x^{\top}=x-\langle x,\nu\rangle\nu.

We claim that

(3.11)

|x^{\top}|\leq c_{0}.

If $x^{\top}=0$ , there is nothing to prove. Otherwise, set $e=x^{\top}/|x^{\top}|.$ Then $e\in T_{x}M$ and $\langle x,e\rangle=|x^{\top}|$ . If $|x^{T}|>c_{0}$ , we would have

\langle x,e\rangle>c_{0}\qquad\text{and}\qquad e\in T_{x}M,

contradicting (3.10). Therefore (3.11) holds. This is also the tangential-position estimate obtained from Alexandrov reflection in [13, Corollary 2.4] and [12, Corollary 19(1)].

We now use the orthogonal decomposition of $x$ . Since $|x|^{2}=\langle x,\nu\rangle^{2}+|x^{\top}|^{2}$ , we have

\displaystyle 1-\left\langle\nu,\frac{x}{|x|}\right\rangle^{2}=\frac{|x|^{2}-\langle x,\nu\rangle^{2}}{|x|^{2}}=\frac{|x^{\top}|^{2}}{|x|^{2}}\leq\left(\frac{c_{0}}{R_{0}-2c_{0}}\right)^{2},

where we have applied (3.11) and Proposition 3.3 for the last inequality.

Notice that, since $3c_{0}<R_{0}$ , then $\frac{c_{0}}{R_{0}-2c_{0}}<1,$ hence the right-hand side of (3.9) is strictly positive. Thus $\left\langle\nu,\frac{x}{|x|}\right\rangle$ does not vanish on $M$ . Since $M$ is connected, this quantity has a constant sign. At a point where $|x|$ attains its minimum, the outward unit normal points away from the origin, because $0\in\Omega$ . Therefore the sign is positive, and $M$ is star-shaped. ∎

Within the class of smooth, closed, connected, embedded hypersurfaces enclosing the same volume as $B_{R_{0}}(0)$ and containing the origin in the enclosed region, strong Alexandrov reflection in every direction and geometric $C^{1}$ -closeness to $\partial B_{R_{0}}(0)$ determine each other quantitatively, up to a change of parameters. More precisely,

Corollary 3.5 (Reflection and geometric $C^{1}$ -closeness).

If $M$ satisfies (1.1) and (1.2), then every $c_{0}>0$ satisfying (3.1) is a reflection threshold for $M$ . Conversely, a common reflection threshold $c_{0}$ , with $3c_{0}<R_{0}$ , implies the radial and normal estimates of Propositions 3.3 and 3.4. In particular, both the relative annular width and the maximal angle between the normal and the radial direction tend to zero as $c_{0}/R_{0}\to 0$ .

Proof.

The first implication is Proposition 3.1. The converse follows from Propositions 3.3 and 3.4. ∎

The importance of the reflection formulation is that it is preserved by the evolution, by Theorem 2.3. Starting from the initial geometric assumptions, we obtain a strong reflection property, propagate it with the same threshold $c_{0}$ , and then apply the converse implication of Corollary 3.5 to each time slice.

3.3. Preservation of the starting conditions under the flow

Proposition 3.6 (Annular closeness and normal oscillation for later times).

Let $M_{0}\subset\mathbb{R}^{n+1}$ satisfy (1.1) and (1.2) and let $M_{t}$ be the solution of the flow (1.3). Let $c_{0}$ be chosen as in Proposition 3.1, and assume that $\varepsilon_{0},\theta_{0}$ are sufficiently small so that $R_{0}>3c_{0}$ .

Then, for every time for which the solution is defined, we have

(3.12)

R_{0}-2c_{0}\leq|x|\leq R_{0}+2c_{0}\qquad\text{on }M_{t}\vskip 6.0pt plus 2.0pt minus 2.0pt

(3.13)

\left\langle\nu,\frac{x}{|x|}\right\rangle^{2}\geq 1-\left(\frac{c_{0}}{R_{0}-2c_{0}}\right)^{2}\qquad\text{on }M_{t}.

Proof.

By Proposition 3.1, $M_{0}$ reflects strictly up to $\Pi_{c_{0},e}$ for every $e\in\mathbb{S}^{n}$ . Then Theorem 2.3 implies that the same threshold $c_{0}$ is preserved throughout the existence interval.

We now wish to apply the static consequences of reflection from the previous subsection. Let us check that the property $0\in\Omega_{t}$ holds throughout the flow. Since $0\in\Omega_{0}$ , let

I=\left\{t\geq 0:0\in\Omega_{s}\text{ for every }0\leq s\leq t\right\}

within the maximal existence interval. This set contains a nontrivial interval starting at $t=0$ . For every $t\in I$ , the hypersurface $M_{t}$ has the same enclosed volume as $M_{0}$ , namely

|\Omega_{t}|=|\Omega_{0}|=\omega_{n+1}R_{0}^{n+1},

and reflects strictly up to $\Pi_{c_{0},e}$ for every direction $e$ . Proposition 3.3 applies to $M_{t}$ with the same constants $R_{0}$ and $c_{0}$ . In particular, the lower radial bound

|x|\geq R_{0}-2c_{0}>0

also shows that $M_{t}$ cannot pass through the origin. Since $0\in\Omega_{0}$ , the origin cannot leave the enclosed region without first belonging to its boundary. Therefore $0\in\Omega_{t}$ for every time in the maximal existence interval. It follows that $I$ is the entire existence interval, and the above estimates hold throughout it.

This shows that we can apply Propositions 3.3 and 3.4 for all $t$ for which the flow exists, for the fixed reference radius $R_{0}$ and the single reflection threshold $c_{0}$ . The conclusion follows. ∎

Corollary 3.7 (Control of the support function).

Under the assumptions of the previous proposition, the support function $u=\langle x,\nu\rangle$ satisfies

(3.14)

R_{0}-3c_{0}\leq u\leq R_{0}+2c_{0}

throughout the existence interval. In particular, setting $\eta_{0}:=3c_{0},$ we have

(3.15)

|u-R_{0}|\leq\eta_{0}

with $\eta_{0}=\eta_{0}(\varepsilon_{0},\theta_{0})\to 0$ as $\varepsilon_{0}+\theta_{0}\to 0$ .

Proof.

First, notice that the tangential estimate (3.11), applied to $x\in M_{t}$ , gives $|x^{\top}|\leq c_{0}.$ Since $x=x^{\top}+u\nu,$ we have $u^{2}=|x|^{2}-|x^{\top}|^{2}.$ Estimate (3.13) gives $u>0$ , while (3.12) yields

u\geq\sqrt{(R_{0}-2c_{0})^{2}-c_{0}^{2}}\geq R_{0}-3c_{0},

where the latter follows from $3c_{0}<R_{0}$ . Finally, the upper bound holds because $u\leq|x|\leq R_{0}+2c_{0}.$ ∎

Remark 3.8.

From now on, we assume that the initial geometric parameters $\varepsilon_{0},\theta_{0}$ are chosen sufficiently small in order that the corresponding reflection threshold satisfies $3c_{0}<R_{0},$ as required in Proposition 3.6. This is certainly possible, in view of Remark 3.2.

4. Curvature control under the flow

In this section we use the $C^{1}$ control from the previous section to obtain a uniform bound of the curvature.

4.1. Control of the mean curvature and the global term

Let us first estimate the global term in the flow equation (1.3).

Lemma 4.1 (Estimate for the nonlocal term).

Let $\eta_{0}=3c_{0}$ be as in Corollary 3.7. If the initial parameters are sufficiently small that $\eta_{0}<\frac{R_{0}}{2},$ then the averaged mean curvature defined in (1.4) satisfies

\bigg|h(t)-\frac{n}{R_{0}}\bigg|\leq\frac{\eta_{0}}{R_{0}}\max_{M_{t}}|H|,

throughout the existence interval.

Proof.

We recall that the support function $u$ satisfies the Minkowski identity

\int_{M}Hu\,d\mu=n|M|.

Using this and the inequality $|u-R_{0}|\leq\eta_{0}$ from Corollary 3.7, we conclude

\bigg|h(t)-\frac{n}{R_{0}}\bigg|=\left|\frac{1}{|M_{t}|R_{0}}\int_{M}H(R_{0}-u)\,d\mu_{t}\right|\leq\frac{\eta_{0}}{R_{0}}\max_{M_{t}}|H|.

∎

Remark 4.2 (The sign of the nonlocal term).

The assumptions of the theorem do not determine the sign of either $H$ or $h$ . Indeed, for a radial graph the second-order part of the mean curvature can be written in divergence form, but its contribution to the total mean curvature does not vanish: the area element introduces a nonconstant weight, and integration by parts leaves a Hessian term with no definite sign. Since our hypotheses control only the radial position and the first derivatives of the graph, this term cannot be estimated from the initial geometric bounds. The Tso-type argument below must therefore treat the different sign configurations of $H$ and $h$ separately, making it more involved than in the usual sign-definite setting.

In the following, when considering some scalar function $f$ defined on $M_{t}$ , we use the notation $f_{\min}:=\min_{M_{t}}f$ and $f_{\max}:=\max_{M_{t}}f$ . Differential inequalities for these quantities are understood in terms of upper and lower Dini derivatives. We shall use the standard scalar maximum principle without further comment.

Proposition 4.3 (Mean curvature is bounded along the flow).

If $\varepsilon_{0},\theta_{0}$ are small enough, then we can find a positive constant $C=C\bigl(n,R_{0},\varepsilon_{0},\theta_{0},\|H(\cdot,0)\|_{L^{\infty}(M_{0})}\bigr)$ satisfying

(4.1)

|H|\leq C\qquad\text{on}\quad M_{t}.

Proof.

Recall that, by Corollary 3.7, after choosing $\varepsilon_{0},\theta_{0}$ sufficiently small, the support function satisfies $|u-R_{0}|\leq\eta_{0}$ , where $\eta_{0}=\eta_{0}(\varepsilon_{0},\theta_{0})$ can be made arbitrarily small.

Setting $\alpha=R_{0}-2\eta_{0}$ , it holds

(4.2)

\eta_{0}\leq u-\alpha\leq 3\eta_{0}.

Next we work with a Tso-type function (cf. [33]), that is, we consider

Z=\frac{H}{u-\alpha}.

We now compute the evolution of $Z$ . The equations from Lemma 2.1 give by direct computation

(4.3)

\displaystyle(\partial_{t}-\Delta)Z=-\frac{\alpha\,Z+h}{u-\alpha}|A|^{2}-\frac{h}{u-\alpha}Z+2Z^{2}+\frac{2}{u-\alpha}\langle\nabla u,\nabla Z\rangle.

We observe that $Z_{\max}$ is always positive, while $Z_{\min}$ can have any sign in general. We want to derive a two-sided bound on $Z$ using the maximum principle. The role of the terms which occur in the equation for $Z$ is different depending on the sign both of $Z$ and of $h(t)$ , therefore we need to treat different cases separately.

Let us first consider a time $t$ at which $h(t)\geq 0$ . At a point where the maximum of $Z$ is attained we can write

\frac{d}{dt}Z_{\max}\leq 2Z_{\max}^{2}-\frac{\alpha\,Z_{\max}}{u-\alpha}|A|^{2}\leq 2Z_{\max}^{2}-\frac{\alpha}{n}Z_{\max}^{3}(u-\alpha)\leq Z_{\max}^{2}\Big(2-\frac{\alpha\,\eta_{0}}{n}Z_{\max}\Big).

We deduce

(4.4)

\frac{d}{dt}Z_{\max}\leq 0\quad\mbox{ if $h(t)\geq 0$ \quad and }Z_{\max}(t)\geq\frac{2n}{\alpha\eta_{0}}.

We now estimate the derivative of $Z_{\min}$ at a time with $h(t)\geq 0$ . Since we want a bound from below, we can suppose that $Z_{\min}<0$ . Using (4.2), we have

(4.5)

|H|_{\max}\leq 3\,\eta_{0}\,|Z|_{\max}<3\,\eta_{0}\,(Z_{\max}-Z_{\min}),

which by Lemma 4.1 yields

h\leq\frac{1}{R_{0}}\Big(n+3\,\eta_{0}^{2}(Z_{\max}-Z_{\min})\Big).

At a point where $Z=Z_{\min}$ , we deduce from (4.3)

\frac{d}{dt}Z_{\min}\geq\frac{|A|^{2}}{u-\alpha}\Big(-\frac{n}{R_{0}}-\frac{3\,\eta_{0}^{2}}{R_{0}}(Z_{\max}-Z_{\min})-\alpha Z_{\min}\Big).

If we choose $\eta_{0}$ sufficiently small so that $\alpha R_{0}-3\eta_{0}^{2}>\frac{\alpha R_{0}}{2}$ , we obtain

\frac{d}{dt}Z_{\min}\geq\frac{|A|^{2}}{(u-\alpha)R_{0}}\Big(-n-3\,\eta_{0}^{2}Z_{\max}+\frac{\alpha R_{0}}{2}(-Z_{\min})\Big).

Then if $-Z_{\min}>\frac{2}{\alpha R_{0}}(n+3\eta_{0}^{2}Z_{\max}(t))$ the right hand side above becomes positive. We conclude that

(4.6)

\frac{d}{dt}Z_{\min}\geq 0\quad\mbox{ if $h(t)\geq 0$ \quad and }Z_{\min}(t)\leq-\frac{2}{\alpha R_{0}}\Big(n+3\eta_{0}^{2}Z_{\max}(t)\Big).

Next, consider a time $t$ at which $h(t)\leq 0$ . We remark that at such a time we have $H_{\min}(t)<0$ and $Z_{\min}(t)<0$ . We first estimate the derivative of $Z_{\min}$ . At a point where $Z=Z_{\min}<0$ , we can discard the positive terms with $-\alpha Z|A|^{2}$ and $Z^{2}$ in the evolution equation (4.3) and obtain

\frac{d}{dt}Z_{\min}\geq-\frac{h}{u-\alpha}\big(|A|^{2}+Z_{\min}\big)\geq-\frac{h}{u-\alpha}\bigg(\frac{H^{2}}{n}+Z_{\min}\bigg)\geq-\frac{hZ_{\min}}{u-\alpha}\bigg(\frac{\eta_{0}^{2}Z_{\min}}{n}+1\bigg).

We deduce

(4.7)

\frac{d}{dt}Z_{\min}\geq 0\quad\mbox{ if $h(t)\leq 0$ \quad and }Z_{\min}(t)\leq-\frac{n}{\eta_{0}^{2}}.

We now consider the derivative of $Z_{\max}$ under the same hypothesis that $h(t)\leq 0$ . Suppose further that $Z_{\max}(t)\geq\frac{n}{\eta_{0}^{2}}>0$ . Then, at a point where $Z=Z_{\max}$ , we have

H=(u-\alpha)Z_{\max}\geq\eta_{0}Z_{\max}\geq\frac{n}{\eta_{0}},

as well as

|A|^{2}\geq\frac{1}{n}H^{2}\geq\frac{u-\alpha}{\eta_{0}}Z_{\max}\geq Z_{\max}.

Accordingly, we can write

	$\displaystyle\frac{d}{dt}Z_{\max}$	$\displaystyle\leq\frac{\|A\|^{2}}{u-\alpha}(-\alpha Z_{\max}-h)-\frac{h}{u-\alpha}Z_{\max}+2Z_{\max}^{2}$
		$\displaystyle\leq\frac{\|A\|^{2}}{u-\alpha}(-\alpha Z_{\max}-h)-\frac{h}{u-\alpha}\|A\|^{2}+2n\frac{\|A\|^{2}}{(u-\alpha)^{2}}$
		$\displaystyle\leq\frac{\|A\|^{2}}{u-\alpha}\bigg(\frac{2n}{\eta_{0}}-2h-\alpha Z_{\max}\bigg).$

By substitution of (4.5) into Lemma 4.1, we obtain

-2h\leq-\frac{2n}{R_{0}}+\frac{6\eta_{0}^{2}}{R_{0}}\left(Z_{\max}-Z_{\min}\right).

If we choose $\eta_{0}$ sufficiently small so that $\alpha R_{0}-6\eta_{0}^{2}>\frac{\alpha R_{0}}{2}$ , we obtain

\displaystyle\frac{d}{dt}Z_{\max}

\displaystyle\leq\frac{|A|^{2}}{u-\alpha}\bigg(\frac{2n}{\eta_{0}}-\frac{2n}{R_{0}}-\frac{\alpha}{2}Z_{\max}-\frac{6\eta_{0}^{2}}{R_{0}}Z_{\min}\bigg).

After discarding the negative term $-2n/R_{0}$ in order to simplify the constants, we get

(4.8)

\frac{d}{dt}Z_{\max}\leq 0\quad\mbox{ if $h(t)\leq 0$ \quad and }Z_{\max}(t)\geq\max\left\{\frac{n}{\eta_{0}^{2}},\frac{2}{\alpha}\left(\frac{2n}{\eta_{0}}-\frac{6\eta_{0}^{2}}{R_{0}}Z_{\min}(t)\right)\right\}.

Let us now define

\gamma_{1}=\max\bigg\{\frac{2n}{\alpha\eta_{0}},\frac{n}{\eta_{0}^{2}},\frac{4n}{\alpha\eta_{0}}\bigg\},\qquad\delta_{1}=\frac{12\eta_{0}^{2}}{\alpha R_{0}}.

\gamma_{2}=\max\bigg\{\frac{2n}{\alpha R_{0}},\frac{n}{\eta_{0}^{2}}\bigg\},\qquad\qquad\delta_{2}=\frac{6\eta_{0}^{2}}{\alpha R_{0}}.

From (4.1), (4.6), (4.7) and (4.8) we see that, regardless of the sign of $h(t)$ , we have

(4.9)

\frac{d}{dt}Z_{\max}\leq 0\quad\mbox{ if }Z_{\max}\geq\gamma_{1}-\delta_{1}[Z_{\min}(t)]_{-}

(4.10)

\frac{d}{dt}Z_{\min}\geq 0\quad\mbox{ if }Z_{\min}\leq-\gamma_{2}-\delta_{2}Z_{\max}(t),

where $[Z_{\min}(t)]_{-}=\min\{Z_{\min}(t),0\}$ .

Since $\eta_{0}$ can be made arbitrarily small by an appropriate choice of the initial parameters $\varepsilon_{0},\theta_{0}$ , we assume that it is chosen in such a way that $\delta_{1}<1,\delta_{2}<1$ . This allows us to take a constant $\Lambda>0$ such that

-\Lambda<Z_{\min}(0)\leq Z_{\max}(0)<\Lambda,

\Lambda>\gamma_{1}+\delta_{1}\Lambda,\qquad-\Lambda<-\gamma_{2}-\delta_{2}\Lambda.

With this choice, we obtain

-\Lambda<Z_{\min}(t)\leq Z_{\max}(t)<\Lambda

for every time for which the solution is defined. In fact, at a first time where either $Z_{\min}(t)=-\Lambda$ , or $Z_{\max}(t)=\Lambda$ , a standard maximum principle argument yields a contradiction with (4.9) or (4.10).

Finally, from (4.2) we reach

|H|=|(u-\alpha)Z|\leq 3\eta_{0}\Lambda,

which proves the uniform bound for the mean curvature. Therefore there exists a constant

C=C\bigl(n,R_{0},\varepsilon_{0},\theta_{0},\|H(\cdot,0)\|_{L^{\infty}(M_{0})}\bigr)

such that $|H|\leq C$ on $M_{t}$ . ∎

Notice that by the lack of convexity assumptions, we cannot conclude that $|A|^{2}$ is bounded from the above uniform control of the mean curvature. We need a separate extra argument inspired by [17] to reach the conclusion.

Proposition 4.4 (Shape operator is bounded along the flow).

Under the same choice of parameters as in Proposition 4.3, there exists a uniform constant bounding $|A|^{2}$ for all times such that the solution is defined.

Proof.

For $x\in M_{t}$ , set

r=|x|,\qquad\omega=\frac{x}{|x|},\qquad q=\langle\nu,\omega\rangle,\qquad v=q^{-1}.

By Proposition 3.6, after choosing $\varepsilon_{0},\theta_{0}$ sufficiently small, there exist constants $0<r_{*}<r^{*}<\infty$ , $\sigma_{*}>0$ , such that along the flow it holds

(4.11)

r_{*}\leq r\leq r^{*},\qquad q=\langle\nu,\omega\rangle\geq\sigma_{*}.\qquad\text{Hence}\qquad 1\leq v\leq v_{*}:=\sigma_{*}^{-1}.

We first record the evolution of $v$ . Since $u=rq$ , the evolution from Lemma 2.1 (b) and

(\partial_{t}-\Delta)r=hq-\frac{n-1+q^{2}}{r}

give

\displaystyle(\partial_{t}-\Delta)q={}

\displaystyle|A|^{2}q+\frac{h}{r}(1-q^{2})-\frac{2H}{r}+\frac{q(n-1+q^{2})}{r^{2}}+\frac{2}{r}\langle\nabla r,\nabla q\rangle.

Since $v=q^{-1}$ , it follows that

(4.12)

(\partial_{t}-\Delta)v=-|A|^{2}v-2v^{-1}|\nabla v|^{2}+\frac{2}{r}\langle\nabla r,\nabla v\rangle+\mathfrak{R},

where

\mathfrak{R}=\frac{2H}{r}v^{2}-\frac{h}{r}(v^{2}-1)-\frac{n-1}{r^{2}}v-\frac{1}{r^{2}v}.

Notice that the last two terms in the definition of $\mathfrak{R}$ are nonpositive. By Proposition 4.3, $|H|+|h|\leq C$ . Together with (4.11), this implies $|\mathfrak{R}|\leq C$ . Hereafter $C$ is a uniform constant depending on initial values, whose concrete meaning may change from line to line.

Choose $k>0$ so that $kv_{*}^{2}<1,$ and define

(4.13)

\mathscr{F}=|A|^{2}\varphi(v^{2}),\qquad\text{with}\qquad\varphi(s)=\frac{s}{1-ks}.

By (4.11), there are positive constants $c_{\varphi},C_{\varphi}$ such that

(4.14)

c_{\varphi}\leq\varphi(v^{2})\leq C_{\varphi}.

Thus a uniform bound for $\mathscr{F}$ is equivalent to a uniform bound for $|A|^{2}$ . Moreover, $\varphi^{\prime}(v^{2})$ and $\varphi^{\prime\prime}(v^{2})$ are uniformly bounded.

Combining (4.12) and Lemma 2.1(c), and carrying out a computation similar to [17, Th. 3.1] while retaining the negative gradient term coming from the evolution of $|A|^{2}$ , we obtain

	$\displaystyle(\partial_{t}-\Delta)\mathscr{F}\leq{}$	$\displaystyle-2\varphi(v^{2})\|\nabla A\|^{2}-2k\mathscr{F}^{2}-\frac{2k}{(1-kv^{2})^{2}}\|\nabla v\|^{2}\mathscr{F}-\frac{2\varphi^{\prime}(v^{2})}{\varphi(v^{2})}\left\langle\nabla v^{2},\nabla\mathscr{F}\right\rangle$
(4.15)			$\displaystyle+\frac{4}{r}\|A\|^{2}v\varphi^{\prime}(v^{2})\langle\nabla r,\nabla v\rangle-2h\,\operatorname{tr}(A^{3})\varphi(v^{2})+2\|A\|^{2}v\varphi^{\prime}(v^{2})\mathfrak{R}.$

We first estimate the term containing $\mathfrak{R}$ . By the uniform boundedness of $\mathfrak{R}$ , $v$ , and $\varphi^{\prime}(v^{2})$ , and by (4.14), we have

(4.16)

\displaystyle 2|A|^{2}v\varphi^{\prime}(v^{2})\mathfrak{R}

\displaystyle\leq C|A|^{2}\leq C\mathscr{F}.

We next rewrite the radial-gradient term as a drift term for $\mathscr{F}$ . Since

\nabla\mathscr{F}=\varphi(v^{2})\nabla|A|^{2}+2|A|^{2}v\varphi^{\prime}(v^{2})\nabla v,

we have

(4.17)

\displaystyle\frac{4}{r}|A|^{2}v\varphi^{\prime}(v^{2})\langle\nabla r,\nabla v\rangle-\frac{2}{r}\langle\nabla r,\nabla\mathscr{F}\rangle=-\frac{2}{r}\varphi(v^{2})\langle\nabla r,\nabla|A|^{2}\rangle.

Using $|\nabla r|\leq 1$ , Kato’s inequality $|\nabla|A|^{2}|\leq 2|A||\nabla A|$ and $r\geq r_{*}>0$ , we obtain, for every $\delta>0$ , by Young’s inequality

(4.18)

\displaystyle\left|\frac{2}{r}\varphi(v^{2})\langle\nabla r,\nabla|A|^{2}\rangle\right|

\displaystyle\leq C\varphi(v^{2})|A||\nabla A|\leq\delta\varphi(v^{2})|\nabla A|^{2}+C_{\delta}\mathscr{F}.

Choose, for instance, $\delta=1$ . Combining (4.17) and (4.18) with the first term in (4.15), we obtain

\displaystyle-2\varphi(v^{2})|\nabla A|^{2}+\frac{4}{r}|A|^{2}v\varphi^{\prime}(v^{2})\langle\nabla r,\nabla v\rangle\leq\frac{2}{r}\langle\nabla r,\nabla\mathscr{F}\rangle+C\mathscr{F},

after removing the non-positive term $-\varphi(v^{2})|\nabla A|^{2}$ . Hence (4.15) and (4.16) imply

(4.19)

\displaystyle(\partial_{t}-\Delta)\mathscr{F}\leq-2k\mathscr{F}^{2}+C\mathscr{F}-\frac{2\varphi^{\prime}(v^{2})}{\varphi(v^{2})}\left\langle\nabla v^{2},\nabla\mathscr{F}\right\rangle+\frac{2}{r}\left\langle\nabla r,\nabla\mathscr{F}\right\rangle-2h\operatorname{tr}(A^{3})\varphi(v^{2}),

where we have discarded the term involving $|\nabla v|^{2}\mathscr{F}$ because it is nonpositive. We still need to control the last cubic term. As $|\operatorname{tr}(A^{3})|\leq|A|^{3}$ and $h$ is uniformly bounded, we get

\displaystyle\left|-2h\,\operatorname{tr}(A^{3})\varphi(v^{2})\right|

\displaystyle\leq C|A|^{3}\varphi(v^{2})=C\mathscr{F}^{3/2}\varphi(v^{2})^{-1/2}\leq C\mathscr{F}^{3/2}\leq\varepsilon\mathscr{F}^{2}+C_{\varepsilon},

which follows by Young’s inequality for every $\varepsilon>0$ . Choosing $0<\varepsilon<k$ , this contribution is absorbed into the negative term $-2k\mathscr{F}^{2}$ and we reach

(4.20)

\displaystyle(\partial_{t}-\Delta)\mathscr{F}\leq{}

\displaystyle-c\mathscr{F}^{2}+C\mathscr{F}+\widetilde{C}-\frac{2\varphi^{\prime}(v^{2})}{\varphi(v^{2})}\left\langle\nabla v^{2},\nabla\mathscr{F}\right\rangle+\frac{2}{r}\left\langle\nabla r,\nabla\mathscr{F}\right\rangle,

for some uniform constants $c>0$ and $C,\widetilde{C}<\infty$ .

At a spatial maximum point of $\mathscr{F}$ , both gradient terms in (4.20) vanish. Therefore,

(4.21)

\frac{d}{dt}\mathscr{F}_{\max}\leq-c\mathscr{F}_{\max}^{2}+C\mathscr{F}_{\max}+\widetilde{C}.

The right-hand side is negative whenever $\mathscr{F}_{\max}$ is sufficiently large. The maximum principle therefore gives a uniform bound for $\mathscr{F}$ . Finally, by (4.14), we conclude

|A|^{2}=\frac{\mathscr{F}}{\varphi(v^{2})}\leq\frac{\mathscr{F}}{c_{\varphi}}.

It follows that $\sup_{M_{t}}|A|^{2}\leq C$ for every time for which the solution is defined.∎

5. Long-time existence and exponential convergence

We complete the proof of Theorem 1.1. Throughout the section, we assume that (1.1) and (1.2) hold, with constants satisfying the same smallness assumptions as in our previous results. We first use the curvature estimate to obtain long-time existence and higher-order regularity.

Proposition 5.1.

The solution of (1.3) exists for every $t\geq 0$ . Moreover, for every integer $m\geq 0$ and every $\tau>0$ , there exists a constant $C_{m,\tau}$ such that

\sup_{M_{t}}|\nabla^{m}A|\leq C_{m,\tau}\qquad\text{for every }t\geq\tau.

Proof.

By Propositions 4.3 and 4.4, we have uniform bounds

|h(t)|\leq\sup_{M_{t}}|H|\leq C,\qquad\sup_{M_{t}}|A|^{2}\leq C

on the maximal time interval of existence.

The evolution equations for the covariant derivatives of the second fundamental form have the usual form

(\partial_{t}-\Delta)\nabla^{m}A=\sum_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+h\sum_{i+j=m}\nabla^{i}A*\nabla^{j}A,

where the symbol $*$ denotes a linear combination of contractions with respect to the induced metric. Since $h=h(t)$ is spatially constant, no spatial derivatives of $h$ occur. The standard Bernstein argument for mean curvature flow therefore gives, for every $m\geq 1$ and every $\tau>0$ ,

\sup_{M_{t}}|\nabla^{m}A|\leq C_{m,\tau},\qquad t\geq\tau;

see, for example, [25, 17]. The constants depend on the uniform bounds for $|A|$ , $h$ , and on $\tau$ .

The continuation criterion for the flow implies that a finite maximal time can occur only if

\limsup_{t\nearrow T_{\max}}\sup_{M_{t}}|A|=\infty.

This is excluded by Proposition 4.4. Hence $T_{\max}=\infty.$ ∎

Proposition 5.2 (Eventual convexity and exponential convergence).

There exist $T<\infty$ and $p_{\infty}\in\mathbb{R}^{n+1}$ such that $M_{t}$ is strictly convex for every $t\geq T$ , and

M_{t}\longrightarrow\partial B_{R_{0}}(p_{\infty})

exponentially in the smooth topology as $t\to\infty$ . In addition, the limit centre point satisfies

(5.1)

|p_{\infty}|\leq\frac{R_{0}}{2}\Phi(\varepsilon_{0},\theta_{0}).

Proof.

The enclosed volume is preserved along the flow, while the area satisfies

(5.2)

\frac{d}{dt}|M_{t}|=-\int_{M_{t}}(H-h)^{2}\,d\mu_{t}.

Since the area is nonincreasing and bounded from below, integration of (5.2) gives

(5.3)

\int_{0}^{\infty}\int_{M_{t}}(H-h)^{2}\,d\mu_{t}\,dt<\infty.

Consequently, there exists a sequence $t_{j}\to\infty$ such that

(5.4)

\int_{M_{t_{j}}}(H-h)^{2}\,d\mu_{t_{j}}\longrightarrow 0.

By Proposition 3.6, the hypersurfaces remain in a fixed compact annulus and are uniformly star-shaped. In particular, they can be written as radial graphs

M_{t}=\left\{r(\vartheta,t)\vartheta:\vartheta\in\mathbb{S}^{n}\right\},

where $r$ is bounded above and below by positive constants and has a uniform gradient bound. Proposition 5.1 and the relations between $A$ and the derivatives of a radial graph then give uniform bounds for all derivatives of $r(\cdot,t)$ , for $t\geq 1$ .

It follows from the Arzelà–Ascoli theorem that, after passing to a subsequence,

r(\cdot,t_{j})\longrightarrow r_{\infty}

smoothly. Thus $M_{t_{j}}$ converges to a smooth, closed, connected and embedded hypersurface $M_{\infty}$ .

The sequence $h(t_{j})$ is bounded. After passing to a further subsequence, we may assume that

h(t_{j})\longrightarrow h_{\infty}.

The smooth convergence and (5.4) imply

H_{M_{\infty}}\equiv h_{\infty}.

Thus $M_{\infty}$ is a closed embedded hypersurface with constant mean curvature. By Alexandrov’s theorem when $n\geq 2$ , and by the classification of closed planar curves with constant curvature when $n=1$ , it is a round sphere.

The $C^{2}$ -convergence to a round sphere implies that $M_{t_{j}}$ is uniformly strictly convex for large enough $j$ . If ${t_{j_{0}}}$ is a time at which our hypersurface is convex, we conclude by Huisken’s theorem [25], that the entire flow $M_{t}$ remains convex for $t\geq{t_{j_{0}}}$ and converges exponentially in the smooth topology to a round sphere as $t\to\infty$ , which necessarily coincides with the limiting sphere $M_{\infty}$ along the subsequence. The corresponding convex convergence result for $n=1$ is due to Gage [20], while the exponential decay follows from Gage’s estimate for the isoperimetric deficit [20], combined with the uniform parabolic estimates for the flow; see also [11].

By the preservation of the enclosed volume, the radius of the limiting sphere is $R_{0}$ . Hence

M_{\infty}=\partial B_{R_{0}}(p_{\infty})

for some $p_{\infty}\in\mathbb{R}^{n+1}$ . Finally, for every sufficiently small $\delta>0$ , Proposition 3.1 applies with $c_{\delta}:=\frac{R_{0}}{2}\Phi(\varepsilon_{0},\theta_{0})+\delta$ . Hence $c_{\delta}$ is a common reflection threshold for $M_{0}$ , and the corresponding radial oscillation estimate (3.7) satisfied by any $M_{t}$ passes to the smooth limit:

2|p_{\infty}|=\max_{M_{\infty}}|x|-\min_{M_{\infty}}|x|\leq 2c_{\delta}

Letting $\delta\searrow 0$ , this concludes the proof. ∎

6. The result in non-flat space forms

6.1. Geometric $C^{1}$ -closeness in space forms

We now turn to the analysis of the volume preserving mean curvature flow (1.3) in space forms. Let $K\in\{-1,1\}$ , and let $N_{K}^{n+1}$ denote the simply connected space form of constant sectional curvature $K$ :

N_{-1}^{n+1}=\mathbb{H}^{n+1},\qquad N_{1}^{n+1}=\mathbb{S}^{n+1}.

We fix a point $o\in N_{K}^{n+1}$ and write

r(x)=\dist_{N_{K}}(o,x).

Let $R_{0}>0$ be determined by

(6.1)

\operatorname{vol}(\Omega_{0})=\operatorname{vol}\bigl(B_{R_{0}}^{K}(o)\bigr),

where $\Omega_{0}$ , as before, is the region enclosed by the initial data. In the spherical case, all hypersurfaces under consideration will be contained in a fixed open hemisphere centred at $o$ . In particular, the reference radius satisfies $0<R_{0}<\frac{\pi}{2}$ .

We use the notation

(6.2)

s_{K}(r)=\begin{cases}\sinh r,&K=-1,\\ \sin r,&K=1,\end{cases}\qquad c_{K}(r)=s_{K}^{\prime}(r).

These functions satisfy $c_{K}^{\prime}=-Ks_{K}$ , $c_{K}^{2}+Ks_{K}^{2}=1$ .

We denote by

\mathbb{S}_{o}^{n}=\left\{v\in T_{o}N_{K}^{n+1}:|v|=1\right\}

the unit sphere in the tangent space at $o$ .

We assume that the initial hypersurface satisfies (1.5) and (1.6) for suitably small values of $\varepsilon_{0}\in(0,1)$ and $\theta_{0}\in(0,\pi/2)$ which will be fixed in the following. We next record the interpretation of these assumptions in radial graph coordinates. Since $\left\langle\nu_{0},\partial_{r}\right\rangle>0,$ the initial hypersurface $M_{0}$ is star-shaped with respect to $o$ , and hence can be written as

M_{0}=\left\{\exp_{o}\bigl(\rho_{0}(\vartheta)\vartheta\bigr):\vartheta\in\mathbb{S}_{o}^{n}\right\}

for a smooth function $\rho_{0}:\mathbb{S}_{o}^{n}\rightarrow(0,\infty).$ In particular, $r\left(\exp_{o}\bigl(\rho_{0}(\vartheta)\vartheta\bigr)\right)=\rho_{0}(\vartheta).$

Let $\sigma$ denote the standard metric on $\mathbb{S}_{o}^{n}$ , and let $\nabla^{\sigma}$ denote its Levi-Civita connection. Set

v_{0}=\sqrt{1+\frac{|\nabla^{\sigma}\rho_{0}|_{\sigma}^{2}}{s_{K}(\rho_{0})^{2}}}.

The outward unit normal is

\nu_{0}=\frac{1}{v_{0}}\left(\partial_{r}-\frac{\sigma^{ij}\partial_{j}\rho_{0}}{s_{K}(\rho_{0})^{2}}\partial_{i}\right).

It follows that

(6.3)

\left\langle\nu_{0},\partial_{r}\right\rangle=\left(1+\frac{|\nabla^{\sigma}\rho_{0}|_{\sigma}^{2}}{s_{K}(\rho_{0})^{2}}\right)^{-1/2}.

Consequently,

(6.4)

\left\langle\nu_{0},\partial_{r}\right\rangle\geq\cos\theta_{0}\quad\Longleftrightarrow\quad|\nabla^{\sigma}\rho_{0}|_{\sigma}\leq s_{K}(\rho_{0})\tan\theta_{0}.

6.2. The initial reflection property

We shall use the standard linear models to represent $N_{K}^{n+1}$ . We set

\mathbb{E}_{K}^{n+2}=\begin{cases}\mathbb{R}^{n+1,1},&K=-1,\\ \mathbb{R}^{n+2},&K=1,\end{cases}

endowed with the Lorentzian scalar product when $K=-1$ and the Euclidean scalar product when $K=1$ . We denote both scalar products by $\langle\cdot,\cdot\rangle_{K}$ . Then

N_{K}^{n+1}=\left\{x\in\mathbb{E}_{K}^{n+2}:\langle x,x\rangle_{K}=K\right\},

where, for $K=-1$ , we take the future component of the hyperboloid. In both cases, we choose our origin $o$ to be the point whose coordinates in the ambient space are $(0,\dots,0,1)$ .

Given $v\in\mathbb{S}_{o}^{n}$ , the unit-speed geodesic issuing from $o$ in the direction $v$ is

(6.5)

\gamma_{v}(s)=c_{K}(s)o+s_{K}(s)v.

Its unit tangent vector is

(6.6)

N_{v,s}:=\dot{\gamma}_{v}(s)=-Ks_{K}(s)o+c_{K}(s)v.

We observe that $\langle\gamma_{v}(s),N_{v,s}\rangle_{K}=0$ . We also set

(6.7)

P_{v,s}=\left\{x\in N_{K}^{n+1}:\langle x,N_{v,s}\rangle_{K}=0\right\}.

Then $P_{v,s}$ is the totally geodesic hypersurface passing through $\gamma_{v}(s)$ and perpendicular to $\gamma_{v}$ . We denote the two components of $N_{K}^{n+1}\setminus P_{v,s}$ by

\mathcal{H}_{v,s}^{+}=\left\{x\in N_{K}^{n+1}:\langle x,N_{v,s}\rangle_{K}>0\right\},\qquad\mathcal{H}_{v,s}^{-}=\left\{x\in N_{K}^{n+1}:\langle x,N_{v,s}\rangle_{K}<0\right\}.

Since

\langle o,o\rangle_{K}=K,\qquad\langle v,v\rangle_{K}=1,\qquad\langle o,v\rangle_{K}=0,

equation (6.6) gives

(6.8)

\langle N_{v,s},N_{v,s}\rangle_{K}=1,\qquad\langle N_{v,s},o\rangle_{K}=-s_{K}(s).

In the following we will only consider $s>0$ . Then we have $\langle o,N_{v,s}\rangle_{K}<0$ , so $o\in\mathcal{H}_{v,s}^{-}$ .

Let us denote by $\mathcal{R}_{v,s}$ the reflection of the ambient linear space across $P_{v,s}$ , that is

(6.9)

\mathcal{R}_{v,s}(x)=x-2\langle x,N_{v,s}\rangle_{K}N_{v,s}.

Since $N_{v,s}$ is a unit spacelike vector, this map preserves $\langle\cdot,\cdot\rangle_{K}$ . It therefore restricts to an isometry of $N_{K}^{n+1}$ , fixes $P_{v,s}$ pointwise and exchanges $\mathcal{H}_{v,s}^{+}$ and $\mathcal{H}_{v,s}^{-}$ . Therefore, $\mathcal{R}_{v,s}$ gives the reflection of $N_{K}^{n+1}$ across the totally geodesic hyperplane $P_{v,s}$ . This explicit representation allowed by the linear model will be convenient in our following analysis.

For fixed $v$ and variable $s$ , the hypersurfaces $P_{v,s}$ are the standard families used in the Alexandrov moving-plane method in space forms [2]. A detailed account of the construction in the hyperbolic case, with its application to geometric flows, has been recently given in [8, Sections 2.1 and 3.1]; for a unified treatment of moving planes in hyperbolic space and in a spherical hemisphere, see [15, Section 2].

For any closed embedded hypersurface $M\subset N_{K}^{n+1}\$ , we can give the definitions regarding the reflection of $M$ exactly as in Definition 2.2, by replacing the Euclidean hyperplanes with the totally geodesic hypersurfaces $P_{v,s}$ . Then Theorem 2.3 by Chow-Gulliver on the preservation of strict reflection along the flow holds unchanged in the space form setting.

We now study the relation between the $C^{1}$ -closeness to a sphere of our initial data and the reflection property. We follow here a different strategy from the one of the Euclidean case: instead of analyzing $M_{0}$ directly as in Proposition 3.1, we first derive estimates for the reflection of a round sphere and then extend the result to $M_{0}$ by a perturbative argument.

Lemma 6.1 (Reflection of the reference sphere).

Let $S_{R_{0}}^{K}(o)=\partial B_{R_{0}}^{K}(o).$ In the spherical case, assume that $R_{0}<\pi/2$ . Then, for every $v\in\mathbb{S}_{o}^{n}$ and $s\in(0,R_{0}]$ , the sphere $S_{R_{0}}^{K}(o)$ reflects strictly across $P_{v,s}$ . Along the reflection hypersurface we have the transversality relation

(6.10)

\left\langle\partial_{r},N_{v,s}\right\rangle_{K}=\frac{s_{K}(s)}{s_{K}(R_{0})}>0\quad\text{on}\quad S_{R_{0}}^{K}(o)\cap P_{v,s}.

In addition, the reflection is uniform for $P_{v,s}$ away from the origin and for $x$ away from $P_{v,s}$ , in the following sense. For any $\gamma>0$ and $\rho>0$ , there exists $\delta_{\gamma,\rho}>0$ such that, for every $s\in[\gamma,R_{0}]$ and $v\in\mathbb{S}^{n}_{o}$ , we have

(6.11)

r\bigl(\mathcal{R}_{v,s}(x)\bigr)\leq R_{0}-\delta_{\gamma,\rho}

for every $x\in S_{R_{0}}^{K}(o)\cap\mathcal{H}_{v,s}^{+}$ such that $\dist(x,P_{v,s})\geq\rho.$

Proof.

If $s=R_{0}$ , then $S_{R_{0}}^{K}(o)$ and $P_{v,s}$ touch at a single point, and the assertions of the theorem are either vacuous or trivial. We then consider $s\in(0,R_{0})$ and take $x\in S_{R_{0}}^{K}(o)\cap\mathcal{H}_{v,s}^{+}.$ We set

a=\langle x,N_{v,s}\rangle_{K},\qquad x^{*}=\mathcal{R}_{v,s}(x),

where $a>0$ since $x\in\mathcal{H}_{v,s}^{+}$ . Observe that, in our model, the distance from $o$ is characterized by

(6.12)

c_{K}(r(x))=K\langle x,o\rangle_{K}.

Since $r(x)=R_{0}$ , equations (6.9) and (6.12) give

(6.13)

c_{K}(r(x^{*}))=K\langle x-2aN_{v,s},o\rangle_{K}=c_{K}(R_{0})-2aK\langle N_{v,s},o\rangle_{K}=c_{K}(R_{0})+2aKs_{K}(s),

which gives $c_{K}(r(x^{*}))-c_{K}(R_{0})=2aKs_{K}(s)$ . Taking into account the different monotonicity of $c_{K}(r)$ for $K=1$ and $K=-1$ , we conclude in both cases that $r(x^{*})<R_{0}$ , that is, $x^{*}$ is contained in the open ball $B_{R_{0}}^{K}(o)$ .

To prove (6.10), let us consider any point $x\in S_{R_{0}}^{K}(o)\cap P_{v,s}$ . From (6.5) and (6.6) we see that there is $\vartheta\in\mathbb{S}_{0}^{n}$ , such that

x=c_{K}(R_{0})o+s_{K}(R_{0})\vartheta

\partial_{r}=-Ks_{K}(R_{0})o+c_{K}(R_{0})\vartheta.

Using $c_{K}^{2}+Ks_{K}^{2}=1$ , these two identities yield

(6.14)

o=c_{K}(R_{0})x-s_{K}(R_{0})\partial_{r}.

Taking the scalar product with $N_{v,s}$ , and using $\langle x,N_{v,s}\rangle_{K}=0$ and (6.8) we obtain

-s_{K}(s)=-s_{K}(R_{0})\langle\partial_{r},N_{v,s}\rangle_{K},

which proves (6.10). That identity implies in particular that $N_{v,s}$ , which is the normal direction to the reflecting hypersurface, is transverse to $S_{R_{0}}^{K}(o)$ at the intersection points; this completes the verification of the strict reflection.

We now prove the uniform radial separation. By rotational symmetry, it suffices to consider a fixed $v\in\mathbb{S}^{n}_{o}$ . For any given $\gamma,\rho>0$ , consider the set

\displaystyle\mathcal{K}_{\gamma,\rho}=\bigl\{(s,x):

\displaystyle\gamma\leq s\leq R_{0},\quad x\in S_{R_{0}}^{K}(o)\cap\mathcal{H}_{v,s}^{+},\quad\dist(x,P_{v,s})\geq\rho\bigr\}.

If $\mathcal{K}_{\gamma,\rho}$ is empty, the assertion is vacuous, and any positive value of $\delta_{\gamma,\rho}$ may be chosen. Otherwise, it is easy to see that $\mathcal{K}_{\gamma,\rho}$ is compact. We know from the first part of the proof that $r(\mathcal{R}_{v,s}(x))<R_{0}$ for all $x\in\mathcal{K}_{\gamma,\rho}$ . Then, setting $\delta_{\gamma,\rho}:=\min_{\mathcal{K}_{\gamma,\rho}}R_{0}-r(\mathcal{R}_{v,s}(x))>0$ , we obtain the sought inequality. ∎

Proposition 6.2 (Initial reflection in space forms).

For any $c_{0}\in(0,R_{0})$ there exist $\varepsilon_{0},\theta_{0}>0$ , only depending on $R_{0},c_{0}$ such that if $M_{0}$ satisfies (1.5)–(1.6) then $M_{0}$ reflects strictly up to $P_{v,c_{0}}$ for every $v\in\mathbb{S}_{o}^{n}$ .

Proof.

Let us fix $c_{0}\in(0,R_{0})$ and take any $s\geq c_{0}$ and $v\in\mathbb{S}_{o}^{n}$ . As before, by rotational symmetry, it is sufficient to take a fixed $v$ . In addition, we only need to consider $s\in[s_{0},R_{0}(1+\varepsilon_{0})]$ since otherwise the outer cap of $M_{0}$ is empty. We will analyze separately points of $M_{0}$ which are close or far from the reflecting hypersurface.

In order to study the behaviour near $P_{v,s}$ , we consider signed Fermi coordinates around $P_{v,s}$ . We denote by $z$ the coordinate corresponding to the signed distance function from $P_{v,s}$ , chosen to be positive on $\mathcal{H}_{v,s}^{+}$ , and by $\partial_{z}$ the corresponding unit tangent vector. Observe that the reflection $\mathcal{R}_{v,s}$ corresponds to reversing the sign $z\mapsto-z$ and leaving the other coordinates unchanged. Observe also that, at points $x\in P_{v,s}$ , we have $\partial_{z}(x)=N_{v,s}$ . We point out that the vector field $\partial_{z}$ depends on $v,s$ (in contrast with the Euclidean case, where it only depends on $v$ ), although we omit writing this dependence explicitly.

We want to show that $\langle\nu,\partial_{z}\rangle>0$ at points of $M_{0}$ close to $P_{v,s}$ . To this purpose, we first consider points on the reference sphere. By (6.10), we have for all $s\in[c_{0},R_{0}]$

\left\langle\partial_{r},\partial_{z}\right\rangle_{K}=\left\langle\partial_{r},N_{v,z}\right\rangle_{K}=\frac{s_{K}(s)}{s_{K}(R_{0})}\geq\frac{s_{K}(c_{0})}{s_{K}(R_{0})}>0\quad\text{on}\quad S_{R_{0}}^{K}(o)\cap P_{v,s}.

By uniform continuity, there exists $\rho>0$ enough small, only depending on $c_{0},R_{0}$ , such that, for all $s\in[c_{0},R_{0}]$ and for all $y\in S_{R_{0}}^{K}(o)$ with $\dist(y,P_{v,s})\leq 3\rho$ , we have

(6.15)

\langle\partial_{r}(y),\partial_{z}(y)\rangle_{K}\geq\frac{3s_{K}(c_{0})}{4s_{K}(R_{0})}.

By the smooth dependence of $\partial_{z}$ on $s$ , after possibly decreasing $\rho>0$ and then $\varepsilon_{0}>0$ , we also have

(6.16)

\left\langle\partial_{r}(x),\partial_{z}(x)\right\rangle_{K}\geq\frac{s_{K}(c_{0})}{2s_{K}(R_{0})}

for all $s\in[c_{0},R_{0}(1+\varepsilon_{0})]$ and every $x\in N^{n+1}_{K}$ for which there exists $y\in S_{R_{0}}^{K}(o)$ satisfying $\dist(x,y)<\rho$ and $\dist(y,P_{v,s})\leq 3\rho$ . Let us additionally require that $\varepsilon_{0}>0$ is small enough to have $R_{0}\varepsilon_{0}\leq\rho$ .

With the above choice of parameters, let us now analyze the reflection of $M_{0}$ across $P_{v,s}$ , for $s\in[c_{0},R_{0}(1+\varepsilon_{0})]$ . We first consider any point $x\in M_{0}$ such that $\dist(x,P_{v,s})\leq 2\rho$ . We can write $x=\exp_{o}\bigl(\rho_{0}(\vartheta)\vartheta\bigr)$ for some $\theta\in\mathbb{S}_{o}^{n}$ . Let $y=\exp_{o}(R_{0}\vartheta)$ be the point in $S_{R_{0}}^{K}(o)$ on the same radial geodesic. By assumption (1.5) we have

\dist(x,y)=|\rho_{0}(\vartheta)-R_{0}|\leq R_{0}\varepsilon_{0}\leq\rho.

Then we also have

\dist(y,P_{v,s})\leq\dist(x,P_{v,s})+\dist(x,y)\leq 3\rho.

By (6.16), we conclude that

\left\langle\partial_{r}(x),\partial_{z}(x)\right\rangle_{K}\geq\frac{s_{K}(c_{0})}{2s_{K}(R_{0})}>0

for all $x\in M_{0}$ such that $\dist(x,P_{v,s})\leq 2\rho$ .

It then follows from (1.6) that for any $x\in M_{0}$ such that $\dist(x,P_{v,s})\leq 2\rho$ , we have

(6.17)

\left\langle\nu_{0}(x),\partial_{z}(x)\right\rangle_{K}\geq\left\langle\partial_{r}(x),\partial_{z}(x)\right\rangle_{K}-|\nu_{0}(x)-\partial_{r}(x)|\geq\frac{s_{K}(c)}{2s_{K}(R_{0})}-\sqrt{2-2\cos\theta_{0}}>0,

provided $\theta_{0}$ is sufficiently small, depending only on $c_{0}$ , $R_{0}$ .

In particular, for points $x\in M_{0}\cap P_{v,s}$ , (6.17) shows that $\langle\nu_{0}(x),N_{v,s}\rangle>0$ , i.e. $N_{v,s}$ is not tangent to $M_{0}$ along the intersection with $P_{v,s}$ .

In addition, similarly to the Euclidean case, the positivity of $\langle\nu_{0},N_{v,s}\rangle$ allows to exclude that a point $x\in M_{0}\cap\mathcal{H}_{v,s}^{+}$ with $\dist(x,P_{v.s})\leq 2\rho$ satisfies $\mathcal{R}_{v,s}(x)\in M_{0}$ . In fact, $x$ and $\mathcal{R}_{v,s}(x)$ lie on the same geodesic perpendicular to $P_{v,s}$ parametrized by $z\in[-2\rho,2\rho]$ . Along such a geodesic, the scalar product $\left\langle\nu_{0},\partial_{z}(x)\right\rangle_{K}$ must change sign on two consecutive transverse intersections with $M_{0}$ , depending on whether the geodesic enters or exits the region enclosed by $M_{0}$ as $z$ increases. On the other hand, (6.17) shows that the sign is always positive. This implies that there can be at most one intersection, and therefore $\mathcal{R}_{v,s}(M_{0})$ cannot intersect $M_{0}$ at points at distance less than $2\rho$ from $P_{v,s}$ .

We observe that, for $R_{0}\leq s\leq R_{0}(1+\varepsilon_{0})$ , every point $x\in M_{0}\cap\mathcal{H}^{+}_{v,s}$ satisfies

\operatorname{dist}(x,P_{v,s})\leq r(x)-s\leq R_{0}\varepsilon_{0}\leq\rho<2\rho.

Thus, for such values of $s$ , the whole outer cap is covered by the preceding near-hyperplane argument.

It remains to consider $s\in[c_{0},R_{0}]$ and points $x\in M_{0}\cap\mathcal{H}^{+}_{v,s}$ such that $\dist(x,P_{v,s})>2\rho$ . As before, we can use (1.5) to show that there is $y\in S_{R_{0}}^{K}(o)$ on the same radial geodesic, which satisfies $\dist(x,y)\leq R_{0}\varepsilon_{0}\leq\rho$ . Then $y$ satisfies

\dist(y,P_{v,s})\geq\dist(x,P_{v,s})-\dist(x,y)\geq\rho

and therefore

r(\mathcal{R}_{v,s}(y))\leq R_{0}-\delta,

where $\delta=\delta(\rho,c_{0},R_{0})>0$ is given from Lemma 6.1. Since $\mathcal{R}_{v,s}$ is an isometry, we deduce

r(\mathcal{R}_{v,s}(x))\leq r(\mathcal{R}_{v,s}(y))+\dist(x,y)\leq R_{0}-\delta+R_{0}\varepsilon_{0}.

For $\varepsilon_{0}>0$ small enough to have $2R_{0}\varepsilon_{0}<\delta$ , we conclude that $r(\mathcal{R}_{v,s}(x))<R_{0}(1-\varepsilon_{0})$ , so that by (1.5) $\mathcal{R}_{v,s}(x)$ lies in the interior of the region bounded by $M_{0}$ .

Thus, we have shown that $M_{0}$ reflects strictly across every $P_{v,s}$ with $v\in S_{o}^{n}$ and $s\geq c_{0}$ . ∎

We next derive the quantitative geometric consequence of the reflection property that will replace the Euclidean estimate for the tangential part of the position vector. Let

(6.18)

Y=s_{K}(r)\partial_{r}

be the radial conformal vector field. Reflection up to distance $c$ in every direction controls the tangential component of $Y$ , and hence the gradient of the radial graph function.

Lemma 6.3 (Gradient estimate from Alexandrov reflection).

Let $M\subset N_{K}^{n+1}$ be a smooth, closed and embedded hypersurface which reflects strictly up to $P_{v,c}$ for every $v\in\mathbb{S}_{o}^{n}$ . Then, wherever $s_{K}(r)>s_{K}(c),$ one has

(6.19)

\left\langle\nu,\partial_{r}\right\rangle\geq\sqrt{1-\frac{s_{K}(c)^{2}}{s_{K}(r)^{2}}}.

If $M$ is written as a radial graph $M=\left\{\exp_{o}\bigl(\rho(\vartheta)\vartheta\bigr):\vartheta\in\mathbb{S}_{o}^{n}\right\},$ then

(6.20)

|\nabla^{\sigma}\rho|_{\sigma}\leq\frac{s_{K}(c)s_{K}(\rho)}{\sqrt{s_{K}(\rho)^{2}-s_{K}(c)^{2}}}.

Proof.

Fix $x\in M$ . If $Y^{T}(x)=0$ , there is nothing to prove. Assume $Y^{T}(x)\neq 0$ , and set $\tau=\frac{Y^{T}(x)}{|Y^{T}(x)|}.$ Then $\langle\partial_{r},\tau\rangle_{K}>0.$

We now use the linear model of $N_{K}^{n+1}$ and consider the hyperplane $\tau^{\perp}\subset\mathbb{E}_{K}^{n+2}$ orthogonal to $\tau$ . As $\tau\in T_{x}M\subset T_{x}N_{K}^{n+1}$ , $x$ belongs to the hypersurface $P_{\tau}=N_{K}^{n+1}\cap\tau^{\perp}$ , which is totally geodesic. Moreover, $\tau\in T_{x}M$ , showing that $M$ does not reflect strictly across $P_{\tau}$ because the transversality condition is violated.

Let $s\geq 0$ denote the distance from $o$ to $P_{\tau}$ . Since $\tau$ is the unit normal to a totally geodesic hypersurface at distance $s$ from $o$ , by (6.8) it holds

s_{K}(s)=-\langle o,\tau\rangle_{K}=s_{K}(r)\langle\partial_{r},\tau\rangle_{K},

where we have written $o=c_{K}(r)x-s_{K}(r)\partial_{r}$ as in the final part of the proof of Lemma 6.1.

Since the reflection property does not hold, the distance of the hypersurface $P_{\tau}$ must satisfy $s<c$ otherwise we contradict our hypotheses. We conclude

\langle\nu,\partial_{r}\rangle^{2}=1-\langle\tau,\partial_{r}\rangle^{2}=1-\frac{s_{K}(s)^{2}}{s_{K}(r)^{2}}\geq 1-\frac{s_{K}(c)^{2}}{s_{K}(r)^{2}}.

The orientation is chosen so that $\langle\nu,\partial_{r}\rangle>0$ , and therefore (6.19) follows. Finally, for a radial graph, combining (6.4) with (6.19) and solving for $|\nabla^{\sigma}\rho|_{\sigma}$ gives (6.20). ∎

We now combine the preceding gradient estimate with volume preservation to control the radial position of the evolving hypersurfaces.

Proposition 6.4 (Radial closeness for later times).

Let $c_{0}>0$ be suitably small (depending on $R_{0}$ ) and let the initial data satisfy (1.5) and (1.6) with $\varepsilon_{0},\theta_{0}>0$ the values given by Proposition 6.2. Then, for every time for which the solution is defined, we have

(6.21)

\big|r(x)-R_{0}\big|\leq 2\pi s_{K}(c_{0})\qquad\text{for every }x\in M_{t}.

In particular, in the case $K=1$ , if $\varepsilon_{0},\theta_{0}$ are suitably small then the solution remains inside the half-sphere of radius $\pi/2$ centred at $o$ .

Proof.

The strict reflection property is preserved by the flow by Theorem 2.3. Then we can apply the gradient estimate from Lemma 6.3 for every time in the existence interval.

In the case $K=1$ , let us set

T^{*}=\sup\left\{T>0:\frac{R_{0}}{2}<r<\frac{\pi}{2}\text{ on }M_{t},\text{ for every }t\in[0,T)\right\}.

In the case $K=-1$ , we define $T^{*}$ analogously, but requiring only the bound from below $\frac{R_{0}}{2}<r$ . The initial annular assumption, together with the condition $R_{0}(1+\epsilon_{0})<\pi/2$ in the $K=1$ case, implies by continuity that $T^{*}>0$ .

By requiring $c_{0}$ to be sufficiently small, we may assume that $s_{K}(c_{0})\leq\sqrt{\frac{3}{4}}s_{K}(R_{0}/2).$ Then on $[0,T^{*})$ estimate (6.20) gives

|\nabla^{\sigma}\rho|_{\sigma}\leq\frac{s_{K}(c_{0})s_{K}(\rho)}{\sqrt{s_{K}(\rho)^{2}-s_{K}(c_{0})^{2}}}\leq 2s_{K}(c_{0}).

Since the diameter of $(\mathbb{S}_{o}^{n},\sigma)$ is $\pi$ , we conclude

(6.22)

\rho_{+}(t)-\rho_{-}(t)\leq 2\pi\,s_{K}(c_{0}),

where $\rho_{-}(t)=\min_{\mathbb{S}_{o}^{n}}\rho(\cdot,t),$ $\rho_{+}(t)=\max_{\mathbb{S}_{o}^{n}}\rho(\cdot,t).$

We now use volume preservation to locate this annulus. Since $\operatorname{vol}(\Omega_{t})=\operatorname{vol}\bigl(B_{R_{0}}^{K}(o)\bigr)$ and $M_{t}$ is a radial graph, the monotonicity of the volume of geodesic balls implies

\rho_{-}(t)\leq R_{0}\leq\rho_{+}(t).

Combining this with (6.22), we obtain

(6.23)

R_{0}-2\pi\,s_{K}(c_{0})\leq\rho(\vartheta,t)\leq R_{0}+2\pi\,s_{K}(c_{0}).

We choose $c_{0}$ so small that $R_{0}-2\pi s_{K}(c_{0})>\frac{R_{0}}{2}$ . In the $K=1$ case, we also require $R_{0}+2\pi\,s_{K}(c_{0})<\pi/2$ . Suppose $T^{*}<T_{\max}$ . Then the smoothness of the flow implies that (6.23) also holds at $t=T^{*}$ . Then we have

\frac{R_{0}}{2}<\min_{M_{T^{*}}}r\leq\max_{M_{T^{*}}}r<\frac{\pi}{2},

where the last inequality only concerns the $K=1$ case. By continuity, the strict inequalities also hold up to a larger time, contradicting the maximality of $T^{*}$ . This shows that $T^{*}=T_{\max}$ and that our radial bound holds for every time for which the solution exists. ∎

Corollary 6.5 (Normal control for later times).

Let our solution satisfy the reflection property with threshold $c_{0}$ . Then, for every time for which it is defined, we have $\langle\nu,\partial_{r}\rangle\geq\sigma_{*}$ , for some $\sigma_{*}=\sigma_{*}(c_{0},R_{0})>0$ . Moreover, for any $R_{0}>0$ , we have $\sigma_{*}(c_{0},R_{0})\to 1$ as $c_{0}\to 0$ .

Proof.

The result follows directly from (6.19) and the lower bound on $r$ given by Proposition 6.4. ∎

Corollary 6.6 (Reflection and geometric $C^{1}$ -closeness in space forms).

Let $K\in\{-1,1\}$ . Within the volume-normalized and hemispherically localized class considered above, strong Alexandrov reflection in every direction and geometric $C^{1}$ -closeness to a geodesic sphere determine each other quantitatively, up to a change of parameters.

More precisely, sufficiently small annular width and normal oscillation give the common reflection threshold of Proposition 6.2. Conversely, a common threshold tending to zero implies the radial and normal estimates of Proposition 6.4 and Corollary 6.5, whose deviations from the reference geodesic sphere also tend to zero.

Proof.

The first implication is Proposition 6.2. For the converse, Lemma 6.3 gives the tangential and normal control from the reflection threshold, while the volume-normalized radial estimate follows from the argument of Proposition 6.4. ∎

The preceding corollary is the space-form counterpart of Corollary 3.5. The difference is that the initial reflection threshold is obtained perturbatively from the strict reflection property of the reference geodesic sphere, rather than from the explicit Euclidean formula (3.1). Once established, however, reflection provides the same type of first-order geometric control required in the curvature estimates.

6.3. Curvature estimates

We now extend the curvature estimates of Section 4 to the non-flat space forms. The proof is the same as in the Euclidean case, except for the presence of ambient-curvature terms. Since $K\in\{-1,1\}$ is constant and the hypersurfaces remain in a fixed compact annulus with a uniform lower bound for $\langle\nu,\partial_{r}\rangle$ , all these additional terms are of lower order and can be absorbed into the constants. We only indicate the necessary modifications.

With the notation from (6.18), the support function reads as $u=\langle Y,\nu\rangle.$ By Proposition 6.4 and Corollary 6.5, there exist constants $r_{*},r^{*}>0$ and $\sigma_{*}>0$ such that

(6.24)

r_{*}\leq r\leq r^{*},\qquad\langle\nu,\partial_{r}\rangle\geq\sigma_{*}

throughout the existence interval. In the spherical case, we also have $r^{*}<\pi/2$ . Consequently,

(6.25)

0<u_{*}\leq u\leq u^{*}<\infty.

The evolution equations of the main geometric quantities take the following form, see, for example, [24, 9, 29].

Lemma 6.7 (Evolution equations in space forms).

Under (1.3) it holds

(a)

$(\partial_{t}-\Delta)H=(H-h)(|A|^{2}+nK)$ .
(b)

$(\partial_{t}-\Delta)|A|^{2}=-2|\nabla A|^{2}+2|A|^{2}\big(|A|^{2}-nK\big)+4KH^{2}-2h\big(\!\operatorname{tr}(A^{3})+KH\big).$
(c)

$(\partial_{t}-\Delta)u=c_{K}(r)(h-2H)+|A|^{2}u.$

Proposition 6.8 (Uniform mean curvature bound).

There exists a constant $C<\infty$ such that

|H|\leq C

for every time for which the solution is defined.

Proof.

Choose $\alpha>0$ such that $u-\alpha\geq\eta_{*}>0$ along the flow, and define

Z=\frac{H}{u-\alpha}.

Using Lemma 6.7 (a) and (c), we obtain

(6.26)

\displaystyle(\partial_{t}-\Delta)Z={}

\displaystyle-\frac{\alpha Z+h}{u-\alpha}|A|^{2}-\frac{h}{u-\alpha}Z+2Z^{2}+\frac{2}{u-\alpha}\langle\nabla u,\nabla Z\rangle+E_{K},

where

E_{K}=nKZ-\frac{nKh}{u-\alpha}-\frac{(c_{K}(r)-1)h}{u-\alpha}Z+2(c_{K}(r)-1)Z^{2},

and the remaining terms on the right-hand side of (6.26) are the same as in the Euclidean calculation. By (6.24) and (6.25), the coefficients in $E_{K}$ are uniformly bounded, hence

(6.27)

|E_{K}|\leq C\bigl(|Z|+Z^{2}+|h|(1+|Z|)\bigr).

Writing $\delta:=\|u-s_{K}(R_{0})\|_{L^{\infty}(M_{t})},$ the space form version of the Minkowski formula gives $|h|\leq C\bigl(1+\delta\|H\|_{L^{\infty}(M_{t})}\bigr)$ , as in Lemma 4.1. By Proposition 6.4 and Corollary 6.5, $\delta$ can be made arbitrarily small by choosing the initial parameters sufficiently small. Hence the part of $h$ which grows with $\|Z\|_{L^{\infty}}$ has an arbitrarily small coefficient, while the remaining contribution is of lower order.

Thus the leading term remains the one involving $Z|A|^{2}$ , because of the estimate

|A|^{2}\geq\frac{H^{2}}{n}=\frac{(u-\alpha)^{2}Z^{2}}{n}.

This term provides the same cubic terms in the upper and lower maximum-principle estimates as in the Euclidean argument, while the additional $K$ -terms are of lower order.

The maximum-principle argument of Proposition 4.3 therefore applies, with only a change in the constants. In this way we obtain uniform bounds from above and below on $Z$ , which in turn yield the desired assertion on $H$ . ∎

Proposition 6.9 (Uniform bound for the shape operator).

There exists a constant $C<\infty$ such that

|A|^{2}\leq C

for every time for which the solution is defined.

Proof.

Setting $q=\langle\nu,\partial_{r}\rangle$ and $v=q^{-1}$ , (6.24) ensures that

1\leq v\leq v_{*}:=\sigma_{*}^{-1}.

The evolution equation for $v$ has the same principal terms as in the Euclidean case:

(6.28)

(\partial_{t}-\Delta)v=-|A|^{2}v-2v^{-1}|\nabla v|^{2}+\langle\mathcal{X},\nabla v\rangle+\mathcal{R}_{K},

where $\mathcal{X}$ is a uniformly bounded tangential vector field and $\mathcal{R}_{K}$ contains the terms involving $H$ , $h$ , the radial coefficients $s_{K}$ , $c_{K}$ , and the curvature $K$ . Notice that no term of order higher than $|A|^{2}v$ is introduced by the ambient curvature. More precisely, $\mathcal{X}$ and $R_{K}$ depend only on the uniformly controlled quantities $r$ , $v$ , $H$ , and $h$ , and on the bounded radial coefficients $s_{K}(r)$ and $c_{K}(r)$ . In particular, they contain no uncontrolled component of $A$ , and hence

|\mathcal{X}|+|\mathcal{R}_{K}|\leq C.

Choose $k>0$ such that $kv_{*}^{2}<1$ , and define $\mathscr{F}$ as in (4.13). As in the Euclidean case, $\mathscr{F}$ is uniformly equivalent to $|A|^{2}$ . Combining the evolution from Lemma 6.7 (b) and (6.28), computing as in the proof of Proposition 4.4, we reach

(6.29)

(\partial_{t}-\Delta)\mathscr{F}\leq-c\mathscr{F}^{2}+C\mathscr{F}+C+\langle\mathcal{X}_{1},\nabla\mathscr{F}\rangle-c_{1}|\nabla v|^{2}\mathscr{F},

for uniform constants $c,c_{1}>0$ and $C<\infty$ . Indeed, the additional terms in Lemma 6.7 (b) satisfy

\left|-2nK|A|^{2}+4KH^{2}-2KhH\right|\leq C(1+|A|^{2}),

and therefore contribute at most $C(\mathscr{F}+1)$ to (6.29). Similarly, the terms collected in $\mathcal{R}_{K}$ contribute at most $C\mathscr{F}$ .

At a spatial maximum of $\mathscr{F}$ , the drift term vanishes and the term involving $|\nabla v|^{2}\mathscr{F}$ is nonpositive. Hence an equation of the form (4.21) also holds in this setting. The maximum principle therefore gives a uniform bound for $\mathscr{F}$ , and hence for $|A|^{2}$ . ∎

Since the ambient space has constant curvature, the evolution equations for the covariant derivatives of $A$ differ from their Euclidean counterparts only by lower-order contractions involving $K$ , $A$ , and its derivatives. The standard Bernstein estimates therefore imply that, for every integer $m\geq 0$ and every $\tau>0$ ,

\sup_{M_{t}}|\nabla^{m}A|\leq C_{m,\tau}\qquad\text{for every }t\geq\tau.

Consequently, the solution exists for every $t\geq 0$ .

6.4. Subsequential convergence

We next use the area dissipation identity and the uniform estimates obtained above to identify the possible limiting hypersurfaces.

Proposition 6.10 (Subsequential convergence).

There exist a sequence $t_{j}\to\infty$ and a point $p_{*}\in N_{K}^{n+1}$ such that

M_{t_{j}}\longrightarrow S_{R_{0}}^{K}(p_{*}):=\partial B_{R_{0}}^{K}(p_{*})

smoothly.

Proof.

The proof is the same as in the first part of Proposition 5.2. Indeed, the area dissipation identity is unchanged, and therefore there exists a sequence $t_{j}\to\infty$ such that (5.4) holds.

The radial and normal estimates, together with the higher-order curvature bounds proved above, give smooth compactness for the radial graph functions

M_{t}=\left\{\exp_{o}\bigl(\rho(\vartheta,t)\vartheta\bigr):\vartheta\in\mathbb{S}_{o}^{n}\right\}.

Thus, after passing to a subsequence, $M_{t_{j}}$ converges smoothly to a closed embedded constant-mean-curvature hypersurface $M_{\infty}$ .

By Alexandrov’s theorem in $N_{K}^{n+1}$ , $M_{\infty}$ is a geodesic sphere. In the spherical case, the fixed-hemisphere assumption ensures that the corresponding version of the theorem applies. Finally, volume preservation and the strict monotonicity of the volume of geodesic balls show that its radius is $R_{0}$ . Hence the statement follows. ∎

6.5. Eventual convexity and exponential convergence

The preceding proposition provides a late time slice which is arbitrarily close in the smooth topology to a geodesic sphere.

We now complete the proof of exponential convergence and at the same time show that convexity is eventually attained, although no convexity or mean-convexity is assumed initially. Smooth subsequential convergence to a geodesic sphere implies that a sufficiently late time slice is uniformly strictly convex. In Euclidean and hyperbolic space, the conclusion then follows by restarting the flow and applying the corresponding convergence theorems for convex hypersurfaces. In the spherical case, where convexity is not preserved in general by (1.3), we instead use the evolution of $|A|^{2}/H^{2}$ . The ambient-curvature term in this evolution has a favourable sign near a geodesic sphere and yields exponential decay of the traceless second fundamental form.

Proposition 6.11 (Eventual convexity and exponential convergence).

Let $K\in\{-1,1\}$ , and assume in the spherical case that $R_{0}<\pi/2$ . Then there exist $T<\infty$ and $p_{\infty}\in N_{K}^{n+1}$ such that $M_{t}$ is strictly convex ( $h$ -convex in case $K=-1$ ) for every $t\geq T$ and

M_{t}\longrightarrow\partial B_{R_{0}}^{K}(p_{\infty})

exponentially in the smooth topology.

Proof.

By Proposition 6.10, there exist a sequence $t_{j}\to\infty$ and $p_{*}\in N_{K}^{n+1}$ such that $M_{t_{j}}\to S_{*}:=\partial B_{R_{0}}^{K}(p_{*})$ smoothly. Setting $\lambda_{0}=\frac{c_{K}(R_{0})}{s_{K}(R_{0})},$ the Weingarten operator of $S_{*}$ is $h_{i}^{j}=\lambda_{0}\delta_{i}^{j}.$ Since $\lambda_{0}>0$ , smooth convergence implies that, for all sufficiently large $j$ ,

\kappa_{i}(\cdot,t_{j})\geq\frac{\lambda_{0}}{2}>0,\qquad i=1,\ldots,n.

Thus $M_{t_{j}}$ is uniformly strictly convex. If $K=-1$ , then $\lambda_{0}=\coth R_{0}>1$ , and we may choose $j$ so large that

\left|\kappa_{i}(\cdot,t_{j})-\coth R_{0}\right|<\frac{\coth R_{0}-1}{2}

for every $i=1,\ldots,n$ . Consequently,

\kappa_{i}(\cdot,t_{j})>\frac{1+\coth R_{0}}{2}>1,\qquad i=1,\ldots,n,

and $M_{t_{j}}$ is strictly convex by horospheres.

Fix one such time $t_{0}=t_{j}$ . If $K=-1$ , the time slice $M_{t_{0}}$ is strictly convex by horospheres. The convergence theorem for (1.3) in hyperbolic space [9] implies that the time-shifted solution remains convex by horospheres and converges exponentially to a geodesic sphere. The enclosed volume is preserved, and therefore the limiting geodesic sphere has radius $R_{0}$ . For the exponential convergence in the hyperbolic plane, one only needs $\kappa(\cdot,t_{j})>0$ and then invoke the corresponding convexity theorem in [34].

It remains to prove the result when $K=1$ . In this case, we could invoke the convergence theorem by Hartley [21]. However, since that result relies on deep arguments using analytic semigroups and centre manifold theory, we provide a direct independent proof, which exploits the stronger properties of the solution obtained above.

We treat separately the cases $n\geq 2$ and $n=1$ . If $n\geq 2$ , we give a direct pinching argument for $Q=\frac{|A|^{2}}{H^{2}}$ .

Since $M_{t_{0}}$ is arbitrarily close to $S_{*}$ in $C^{\infty}$ , after increasing $t_{0}$ we may assume that

(6.30)

\frac{\lambda_{0}}{2}\leq\kappa_{i}\leq 2\lambda_{0},\qquad\frac{n\lambda_{0}}{2}\leq H,h\leq 2n\lambda_{0},\qquad\frac{h}{H}\leq\frac{3}{2}

on $M_{t_{0}}$ , where $\lambda_{0}=\cot R_{0}$ .

By Lemma 6.7 (a) and (b) (see also [29, Lemma 5.1]), one has

	$\displaystyle(\partial_{t}-\Delta)Q={}$	$\displaystyle\frac{2}{H}\langle\nabla H,\nabla Q\rangle-\frac{2}{H^{4}}\|H\nabla A-A\otimes\nabla H\|^{2}$
(6.31)			$\displaystyle+\frac{2h}{H^{3}}\bigl(\|A\|^{4}-H\operatorname{tr}(A^{3})\bigr)-2n\left(2-\frac{h}{H}\right)\left(Q-\frac{1}{n}\right).$

Indeed, the $K$ -terms in the quotient evolution for $K=1$ are

-\frac{2n|A|^{2}}{H^{2}}+4-\frac{2h}{H}-\frac{2n|A|^{2}(H-h)}{H^{3}}=2(1-nQ)\left(2-\frac{h}{H}\right).

Moreover, since $Q-\frac{1}{n}=\frac{|\mathring{A}|^{2}}{H^{2}}$ , if $\kappa_{i}\geq m>0$ , then

H\operatorname{tr}(A^{3})-|A|^{4}=\sum_{i<j}\kappa_{i}\kappa_{j}(\kappa_{i}-\kappa_{j})^{2}\geq m^{2}\sum_{i<j}(\kappa_{i}-\kappa_{j})^{2}=n\,m^{2}|\mathring{A}|^{2}.

Consequently,

\frac{2h}{H^{3}}\bigl(|A|^{4}-H\operatorname{tr}(A^{3})\bigr)\leq-\frac{2nhm^{2}}{H}\left(Q-\frac{1}{n}\right).

Let $T^{*}>t_{0}$ be the supremum of the times for which

\frac{\lambda_{0}}{4}<\kappa_{i}<3\lambda_{0},\qquad\frac{n\lambda_{0}}{4}<H,h<3n\lambda_{0},\qquad\frac{h}{H}<\frac{7}{4}

holds on $[t_{0},T^{*})$ . On this interval we may take $m=\lambda_{0}/4$ , and the preceding estimates yield

(6.32)

(\partial_{t}-\Delta)Q\leq\frac{2}{H}\langle\nabla H,\nabla Q\rangle-\mu_{0}\left(Q-\frac{1}{n}\right)

for a constant $\mu_{0}=\mu_{0}(n,R_{0})>0$ . At a spatial maximum of $Q$ , the drift term vanishes. Hence the maximum principle gives

(6.33)

\max_{M_{t}}\frac{|\mathring{A}|^{2}}{H^{2}}\leq e^{-\mu_{0}(t-t_{0})}\max_{M_{t_{0}}}\frac{|\mathring{A}|^{2}}{H^{2}}\qquad\text{for every \quad$t<T^{*}$.}

Since $H$ is uniformly bounded above and below on this interval, the latter implies

(6.34)

\|\mathring{A}(\cdot,t)\|_{L^{\infty}}\leq Ce^{-\gamma(t-t_{0})}

for some $\gamma>0$ . The uniform higher-order curvature estimates proved above, together with interpolation, give for every $m\geq 0$

(6.35)

\|\nabla^{m}\mathring{A}(\cdot,t)\|_{L^{\infty}}\leq C_{m}e^{-\gamma_{m}(t-t_{0})}.

This, by means of the Codazzi identity $\nabla^{i}\mathring{h}_{ij}=\frac{n-1}{n}\nabla_{j}H,$ gives $\|\nabla H(\cdot,t)\|_{L^{\infty}}\leq Ce^{-\gamma(t-t_{0})}$ .

By Proposition 6.4 and Corollary 6.5, the radial graph functions satisfy

0<\rho_{*}\leq\rho(\cdot,t)\leq\rho^{*},\qquad|\nabla^{\sigma}\rho(\cdot,t)|_{\sigma}=s_{K}(\rho)\sqrt{\left\langle\nu,\partial_{r}\right\rangle^{-2}-1}\leq C.

Since the induced metric is $g_{t}=d\rho\otimes d\rho+s_{K}(\rho)^{2}\sigma,$ it follows that $g_{t}\leq C\sigma$ , and hence $\operatorname{diam}(M_{t},g_{t})\leq C.$ Consequently,

\operatorname{osc}_{M_{t}}H\leq C\|\nabla H(\cdot,t)\|_{L^{\infty}}\leq Ce^{-\gamma(t-t_{0})}.

Since $H_{\min}\leq h\leq H_{\max}$ , we obtain $\|H-h\|_{L^{\infty}(M_{t})}\leq Ce^{-\gamma(t-t_{0})}$ . Using (6.35), the evolution equations and parabolic interpolation once more, this improves to

(6.36)

\|H-h\|_{C^{m}(M_{t})}\leq C_{m}e^{-\gamma_{m}(t-t_{0})}

for every $m\geq 0$ . Using the radial graph parametrization fixed above, by the flow equation (1.3), the latter implies

\int_{t_{0}}^{T^{*}}\|\partial_{t}X(\cdot,t)\|_{C^{m}}\,dt\leq C_{m}\|\mathring{A}(\cdot,t_{0})\|_{C^{m+1}}.

Choose $t_{0}=t_{j}$ sufficiently large that $M_{t_{0}}$ belongs to a smaller neighbourhood whose closure is contained in the bootstrap region defining $T^{*}$ . The estimates above, with their dependence on the initial pinching made explicit, give

\|\partial_{t}X(\cdot,t)\|_{C^{m}}\leq C_{m}\varepsilon_{*}e^{-\gamma_{m}(t-t_{0})},

where $\varepsilon_{*}=\|\mathring{A}(\cdot,t_{0})\|_{C^{m+1}}+\|H(\cdot,t_{0})-h(t_{0})\|_{C^{m}}\to 0$ as $t_{0}=t_{j}\to\infty$ . Hence

\|X(\cdot,t)-X(\cdot,t_{0})\|_{C^{m}}\leq\frac{C_{m}}{\gamma_{m}}\varepsilon_{*}\qquad\text{for every }t<T^{*}.

Taking $t_{0}$ sufficiently large, the solution remains at a uniform positive distance from the boundary of the bootstrap region. Here the bootstrap neighbourhood is understood in a fixed $C^{2}$ -neighbourhood of $S_{R_{0}}^{1}(p_{*})$ ; membership in this neighbourhood controls $\kappa_{i}$ , $H$ , $h$ , and $h/H$ .

In particular, none of the inequalities defining $T^{*}$ can become an equality. If $T^{*}<\infty$ , continuity would then allow the bootstrap conditions to be extended beyond $T^{*}$ , contradicting its definition. Therefore $T^{*}=\infty$ . In particular, $M_{t}$ remains strictly convex for every $t\geq t_{0}$ .

Equation (6.36) also shows that $X(\cdot,t)$ is Cauchy in every $C^{m}$ -norm. Thus it converges exponentially to a smooth limiting embedding $X_{\infty}$ . By (6.34), the limiting hypersurface satisfies $\mathring{A}_{\infty}=0$ . Hence it is totally umbilical. Since it is closed, embedded and contained in an open hemisphere, it is a geodesic sphere $M_{\infty}=\partial B_{R_{\infty}}^{1}(p_{\infty})$ . The enclosed volume is preserved, so $\operatorname{vol}\bigl(B_{R_{\infty}}^{1}(p_{\infty})\bigr)=\operatorname{vol}\bigl(B_{R_{0}}^{1}(o)\bigr)$ , and therefore $R_{\infty}=R_{0}$ . This concludes the proof when $n\geq 2$ .

Let us now consider the case of curves. As before, by Proposition 6.10 we have the smooth subsequential convergence

M_{t_{j}}\longrightarrow S_{*}:=\partial B_{R_{0}}^{1}(p_{*}).

We first show that the whole flow converges smoothly to $S_{*}$ . Taking $p_{*}$ as the reference point, let $\varepsilon_{j}$ and $\theta_{j}$ denote the annular and radial-tilt parameters of $M_{t_{j}}$ . The smooth convergence above gives $\varepsilon_{j}+\theta_{j}\longrightarrow 0$ . As, by Proposition 6.2, every fixed $c>0$ is a common reflection threshold for $M_{t_{j}}$ when $j$ is sufficiently large, we may choose common reflection thresholds $c_{j}\to 0$ .

By Proposition 6.4, the reflection threshold $c_{j}$ is preserved for every $t\geq t_{j}$ . The quantitative converse in Corollary 6.6, applied uniformly on the whole time interval $[t_{j},\infty)$ , therefore gives

\sup_{t\geq t_{j}}\bigl\|\rho_{p_{*}}(\cdot,t)-R_{0}\bigr\|_{C^{1}}\longrightarrow 0,

where $\rho_{p_{*}}(\cdot,t)$ denotes the radial graph function of $M_{t}$ with respect to $p_{*}$ . Since the estimate holds on the entire tail $[t_{j},\infty)$ , it yields

\rho_{p_{*}}(\cdot,t)\longrightarrow R_{0}\qquad\text{in }C^{1}\text{ as }t\to\infty,

without passing to a subsequence. The uniform higher-order estimates and interpolation then upgrade this convergence to every $C^{m}$ norm. Hence $M_{t}\to S_{*}$ smoothly as $t\to\infty$ .

It remains to prove the exponential rate. By the definition of $h$ ,

(6.37)

\int_{M_{t}}w\,ds=0\qquad\text{for}\quad w:=\kappa-h.

Moreover, if $Y$ is an ambient Killing field on $\mathbb{S}^{2}$ , the one-parameter group of isometries generated by $Y$ preserves both length and enclosed area. The corresponding first-variation formulas give

(6.38)

\int_{M_{t}}w\langle Y,\nu\rangle\,ds=0.

On the other hand, the normal component of every Killing field $Y$ satisfies

-\partial_{s}^{2}\langle Y,\nu\rangle=\bigl(\kappa^{2}_{S_{\ast}}+\operatorname{Sec}_{\mathbb{S}^{2}}\bigr)\langle Y,\nu\rangle=(\cot^{2}R_{0}+1)\langle Y,\nu\rangle=\frac{1}{\sin^{2}R_{0}}\langle Y,\nu\rangle;

see, for instance, [19, Proposition 1]. The normal components of the Killing fields obtained by moving the centre of $S_{*}$ in two independent directions are, up to a choice of origin and normalization, the sine and cosine functions on $S_{*}$ . Hence, on $M_{t}$ , by (6.37) and (6.38), $w$ is orthogonal to the constants and to a two-dimensional space converging smoothly to this first eigenspace. Since $S_{*}$ is intrinsically a circle of radius $\sin R_{0}$ , the next eigenvalue of $-\partial_{s}^{2}$ is $4\sin^{-2}R_{0}$ ; see [6, Chapter II.4]. The Rayleigh characterization (cf. [10, Chapter I.5]) and the smooth convergence $M_{t}\to S_{*}$ therefore imply that, for every $\varepsilon>0$ and all sufficiently large $t$ ,

\int_{M_{t}}|\partial_{s}w|^{2}\,ds\geq\left(4a_{0}-\varepsilon\right)\int_{M_{t}}w^{2}\,ds,\qquad\text{with}\quad a_{0}:=\frac{1}{\sin^{2}R_{0}}.

Now set $E(t):=\int_{M_{t}}w^{2}\,ds$ . The evolution equations $\partial_{t}\kappa=\partial_{s}^{2}\kappa+(\kappa-h)(\kappa^{2}+1)$ and $\partial_{t}ds=-\kappa(\kappa-h)\,ds$ yield

E^{\prime}(t)=-2\int_{M_{t}}|\partial_{s}w|^{2}\,ds+2\int_{M_{t}}(\kappa^{2}+1)w^{2}\,ds-\int_{M_{t}}\kappa w^{3}\,ds,

where the term containing $h^{\prime}(t)$ vanishes because of (6.37).

Since $M_{t}\to S_{*}$ smoothly, for every $\varepsilon>0$ and all sufficiently large $t$ , one can guarantee

|\kappa^{2}+1-a_{0}|\leq\varepsilon,\qquad\|w\|_{L^{\infty}(M_{t})}\leq\varepsilon,\qquad\|\kappa\|_{L^{\infty}(M_{t})}\leq C_{0},

where $C_{0}$ is independent of $t$ . Applying the preceding estimates, we obtain

E^{\prime}(t)\leq-2(4a_{0}-\varepsilon)E(t)+2(a_{0}+\varepsilon)E(t)+C_{0}\varepsilon E(t)\leq-3a_{0}E(t),

after choosing $\varepsilon>0$ so that $(4+C_{0})\varepsilon\leq 3a_{0}.$ Therefore, for all sufficiently large $t$ ,

E(t)\leq Ce^{-3a_{0}t}.

The uniform higher-order curvature estimates and interpolation now give, for every integer $m\geq 0$ , constants $C_{m},\gamma_{m}>0$ such that

\|\kappa(\cdot,t)-h(t)\|_{C^{m}(M_{t})}\leq C_{m}e^{-\gamma_{m}t}.

Finally, in radial coordinates with respect to $p_{*}$ , $\partial_{t}\rho_{p_{*}}=\frac{h-\kappa}{\langle\nu,\partial_{r}\rangle}.$ The denominator is uniformly bounded away from zero, and its derivatives are uniformly controlled. Since $\rho_{p_{*}}(\cdot,t)\to R_{0}$ smoothly, integration from $t$ to infinity gives

\|\rho_{p_{*}}(\cdot,t)-R_{0}\|_{C^{m}}\leq C_{m}e^{-\gamma_{m}t}.

Thus $M_{t}$ converges exponentially in the smooth topology to $S_{*}$ . ∎

Corollary 6.12 (Location of the limiting centre).

Let $c_{0}$ be as in Proposition 6.2. Then

\dist_{N_{K}}(o,p_{\infty})\leq c_{0}.

In particular, $\dist_{N_{K}}(o,p_{\infty})\to 0$ as $\varepsilon_{0}+\theta_{0}\to 0$ .

Proof.

As in the Euclidean case, the preserved reflection property passes to the smooth limit. Hence the limiting ball $B_{R_{0}}^{K}(p_{\infty})$ reflects inside itself across every $P_{v,s}$ with $v\in\mathbb{S}_{o}^{n}$ and $s\geq c_{0}$ . This easily implies that the centre of the ball has distance at most $c_{0}$ from $o$ . Since Proposition 6.2 implies the reflection property for every prescribed $c_{0}>0$ , provided that $\varepsilon_{0}$ and $\theta_{0}$ are sufficiently small, the last assertion follows. ∎

7. Extension to Lipschitz initial data

The annular-closeness and normal-oscillation assumptions make sense for radial graphs of merely Lipschitz regularity. We prove that sufficiently small such data admit an instantaneously smooth volume preserving mean curvature flow. The construction is geometric: smooth approximations inherit a common reflection threshold, and the weighted curvature estimate gives bounds independent of their initial curvature.

In the following statement, we define $s_{K}(\cdot)$ as in (6.2) when $K=\pm 1$ , and $s_{K}(\rho)=\rho$ for $K=0$ .

Theorem 7.1 (Lipschitz stability in space forms).

Let $K\in\{-1,0,1\}$ , and let

M_{0}=\left\{\exp_{o}\bigl(\rho_{0}(\vartheta)\vartheta\bigr):\vartheta\in\mathbb{S}_{o}^{n}\right\}\subset N_{K}^{n+1},\qquad\text{with}\qquad\rho_{0}\in W^{1,\infty}(\mathbb{S}_{o}^{n}).

Assume that

(7.1)

R_{0}(1-\varepsilon_{0})\leq\rho_{0}\leq R_{0}(1+\varepsilon_{0})\qquad\text{and}\qquad\frac{|\nabla^{\sigma}\rho_{0}|_{\sigma}}{s_{K}(\rho_{0})}\leq\tan\theta_{0}\qquad\text{a.e. on }\mathbb{S}_{o}^{n}.

When $K=1$ , require in addition that $R_{0}(1+\varepsilon_{0})<\frac{\pi}{2}.$ If $\varepsilon_{0}$ and $\theta_{0}$ are sufficiently small, then

(a)

There exists a family of radial graphs $M_{t}=\left\{\exp_{o}\bigl(\rho(\vartheta,t)\vartheta\bigr):\vartheta\in\mathbb{S}_{o}^{n}\right\},\,t>0,$ which solves (1.3) classically for every $t>0$ and is smooth for every positive time. If $\Omega_{t}$ denotes the region enclosed by $M_{t}$ , then $\operatorname{vol}(\Omega_{t})=\operatorname{vol}(\Omega_{0})\,\text{for every }t>0.$ Moreover, for all sufficiently small $t>0$ ,

$\|\rho(\cdot,t)-\rho_{0}\|_{L^{\infty}(\mathbb{S}_{o}^{n})}\leq C\sqrt{t}.$

In particular, $\rho(\cdot,t)\rightarrow\rho_{0}$ uniformly as $t\searrow 0$ , and $M_{t}$ converges to $M_{0}$ in the Hausdorff sense.
(b)

For every $T>0$ and every integer $m\geq 0$ , there exists $C_{m,T}<\infty$ such that

$\sup_{0<t\leq T}t^{(m+1)/2}\|\nabla^{m}A(\cdot,t)\|_{L^{\infty}(M_{t})}\leq C_{m,T}.$
(c)

The solution exists for all times and satisfies the same conclusions as in Theorems 1.1 and 1.2 as $t\to\infty$ . In particular, it converges to a geodesic sphere, exponentially in the $C^{\infty}$ topology.

By Rademacher’s theorem, the gradient in (7.1) can be equivalently interpreted as the classical gradient at points of differentiability, or in the weak sense. By (6.3), we see that the requirement on $\nabla^{\sigma}\rho_{0}$ in (7.1) is equivalent to the condition $\langle\nu_{0},\partial_{r}\rangle\geq\cos\theta$ in (1.2) and (1.6). Observe that one could give an equivalent formulation in terms of the smallness of the Lipschitz constant of $\rho(\cdot)$ (depending on $R_{0}$ ).

Proof.

The construction proceeds by smooth approximation. We give the complete argument in Euclidean space, and in the end indicate the modifications required when $K=\pm 1$ .

In order to approximate smoothly our initial datum $\rho_{0}\in W^{1,\infty}(\mathbb{S}^{n})$ in a way which preserves the bounds in (7.1), it is convenient to work first with the function

(7.2)

w_{0}=\log\frac{\rho_{0}}{R_{0}},

which satisfies

\log(1-\varepsilon_{0})\leq w_{0}\leq\log(1+\varepsilon_{0})\qquad\text{and }\qquad|\nabla^{\sigma}w_{0}|_{\sigma}\leq\tan\theta_{0}\qquad\text{a.e.}

Let $P_{t}=e^{t\Delta_{\sigma}}$ denote the heat semigroup on $\mathbb{S}^{n}$ , and choose any sequence $t_{j}\searrow 0$ . Set

\widetilde{w}_{0}^{(j)}=P_{t_{j}}w_{0}.

Then the functions $\widetilde{w}_{0}^{(j)}$ are in $C^{\infty}(\mathbb{S}^{n})$ , satisfy $\log(1-\varepsilon_{0})\leq\widetilde{w}_{0}^{(j)}\leq\log(1+\varepsilon_{0})$ , and converge to $w_{0}$ uniformly on $\mathbb{S}^{n},$ since $w_{0}$ is Lipschitz. Moreover, as $\operatorname{Ric}_{\sigma}=(n-1)\sigma$ , well known properties of the heat kernel (cf. [32, Theorem 1.3 and Corollary 1.4]) imply that

\|\nabla^{\sigma}\widetilde{w}_{0}^{(j)}\|_{L^{\infty}}\leq e^{-(n-1)t_{j}}\|\nabla^{\sigma}w_{0}\|_{L^{\infty}}\leq\tan\theta_{0}.

We then define the functions $\widetilde{\rho}_{0}^{(j)}=R_{0}e^{\widetilde{w}_{0}^{(j)}},$ which are smooth approximants of $\rho_{0}$ . We multiply them by a factor $a_{j}>0$ , chosen so that $\rho_{0}^{(j)}:=a_{j}\widetilde{\rho}_{0}^{(j)}$ encloses the same volume as $M_{0}$ . Then it is easy to check that $a_{j}\to 1$ , and that the functions $\rho_{0}^{(j)}$ converge uniformly to $\rho_{0}$ . In addition, they satisfy

(7.3)

R_{0}(1-\varepsilon_{0}-\delta_{j})\leq\rho_{0}^{(j)}\leq R_{0}(1+\varepsilon_{0}+\delta_{j})

for suitable numbers $\delta_{j}\searrow 0$ and, for every sufficiently large $j$ ,

(7.4)

\big|\nabla^{\sigma}\log\rho_{0}^{(j)}\big|_{\sigma}\leq\tan\theta_{0}.

For each $j\geq 1$ , we now solve the flow (1.3) with the smooth initial data $M_{0}^{(j)}$ and denote by $M_{t}^{(j)},$ the corresponding smooth solutions of (1.3). Using properties (7.3) and (7.4), we can apply our previous results to obtain uniform $C^{1}$ -bounds on the solutions.

In fact, let $c_{0}$ be the reflection threshold fixed in Remark 3.8. As inequality (3.1) is strict, from Proposition 3.1 we obtain that $c_{0}$ is a common reflection threshold for $M_{0}^{(j)}$ for every sufficiently large $j$ . Then we can apply the subsequent results of Section 3, to show that the reflection property is preserved along the flow, and the solutions satisfy, for every $j$ and $t$ ,

(7.5)

R_{0}-2c_{0}\leq|x^{(j)}|\leq R_{0}+2c_{0},

(7.6)

q^{(j)}:=\left\langle\nu^{(j)},\frac{x^{(j)}}{|x^{(j)}|}\right\rangle\geq q_{*}(c_{0}):=\left(1-\frac{c_{0}^{2}}{(R_{0}-2c_{0})^{2}}\right)^{1/2}>0.

By the results of Sections 4 and 5, we obtain that each $M_{t}^{(j)}$ exists globally and converges to a sphere. We want to show that these solutions converge, as $j\to\infty$ , to a limit flow $M_{t}$ which satisfies the conclusions of our theorem. To this purpose, the crucial step is to obtain uniform curvature bounds on $M_{t}^{(j)}$ .

To estimate the curvature, we will adapt the strategy of Section 4 in a way that the bounds are independent of the curvature of the initial data. This time we no longer prove a separate bound on $H$ as in Proposition 4.3; instead, we proceed directly to estimating $|A|^{2}$ by a suitable adaptation of the technique of Proposition 4.4.

We first observe that, by the choice of $c_{0}$ in Remark 3.8, the constant in (7.6) satisfies $q_{*}(c_{0})\to 1$ as $c_{0}/R_{0}\to 0$ . We may therefore fix $k$ and choose the initial geometric parameters sufficiently small that

(7.7)

\frac{\sqrt{n}}{1+\sqrt{n}}<k<q_{*}(c_{0})^{2}.

Set $v=(q^{(j)})^{-1}$ and define $\mathscr{F}$ as in (4.13).

By (7.6), it holds $1\leq v\leq q_{*}(c_{0})^{-1}$ and hence $kv^{2}<1$ . Thus $\mathscr{F}$ and $|A|^{2}$ are uniformly equivalent; in particular, $\mathscr{F}\geq\varphi(1)|A|^{2}$ .

Now let $\mathscr{G}_{j}(t):=\max_{M_{t}^{(j)}}F$ , and compute hereafter at a point where the maximum is attained. Arguing as in the proof of Proposition 4.4, but without using any bound for $|h|$ and $|H|$ , we obtain

(7.8)

\displaystyle D^{+}\mathscr{G}_{j}\leq-2k\mathscr{G}_{j}^{2}+C(1+|h|+|H|)\mathscr{G}_{j}-2h\operatorname{tr}(A^{3})\varphi(v^{2}),

in the upper Dini-derivative sense. Here $C$ depends only on $n$ , $R_{0}$ , and the uniform first-order bounds in (7.5) and (7.6). In particular, $C$ is independent of $j$ and of the curvature of $M_{0}^{(j)}$ .

For every $t\in(0,T_{j})$ ,

(7.9)

\displaystyle|h(t)|

\displaystyle\leq\|H(\cdot,t)\|_{L^{\infty}(M_{t}^{(j)})}\leq\sqrt{n}\,\|A(\cdot,t)\|_{L^{\infty}(M_{t}^{(j)})}\leq\sqrt{n(1-k)}\,\mathscr{G}_{j}(t)^{1/2}.

Then the last term in (7.8), which has an indefinite sign, can be estimated as follows:

(7.10)

\displaystyle 2|h|\,|\operatorname{tr}(A^{3})|\varphi(v^{2})

\displaystyle\leq 2\sqrt{n(1-k)}\,\mathscr{G}_{j}^{1/2}\frac{\mathscr{G}_{j}^{3/2}}{\sqrt{\varphi(1)}}=2\sqrt{n}(1-k)\mathcal{G}_{j}^{2}.

Combining the above estimates, we reach

(7.11)

\displaystyle D^{+}\mathscr{G}_{j}\leq{}

\displaystyle-2\bigl(k-\sqrt{n}(1-k)\bigr)\mathscr{G}_{j}^{2}+C\mathscr{G}_{j}(1+C\mathscr{G}_{j}^{1/2}).

By (7.7), we can choose $0<\eta<k-\sqrt{n}(1-k)$ so that Young’s inequality gives $C\mathscr{G}_{j}^{3/2}\leq 2\eta\mathscr{G}_{j}^{2}+C_{\eta}.$ Consequently, there exist constants $\mu>0$ and $C<\infty$ , independent of $j$ , such that

(7.12)

D^{+}\mathscr{G}_{j}\leq-\mu\mathscr{G}_{j}^{2}+C(\mathscr{G}_{j}+1).

Choose $M_{*}>0$ , independently of $j$ , such that $C(s+1)\leq\frac{\mu}{2}s^{2}$ for every $s\geq M_{*}$ . Then we get $D^{+}\mathscr{G}_{j}\leq-\frac{\mu}{2}\mathscr{G}_{j}^{2}$ whenever $\mathscr{G}_{j}\geq M_{*}.$ A comparison argument gives $\mathscr{G}_{j}(t)\leq\max\left\{M_{*},\frac{2}{\mu t}\right\}$ for $0<t<T_{j}$ . Consequently, we conclude that

(7.13)

\sup_{M_{t}^{(j)}}|A|^{2}\leq C\left(1+\frac{1}{t}\right),\qquad t>0

In particular,

(7.14)

\|H(\cdot,t)\|_{L^{\infty}(M_{t}^{(j)})}+|h(t)|\leq C\left(1+\frac{1}{\sqrt{t}}\right),\qquad t>0.

The evolution equations for the covariant derivatives of $A$ , together with the standard interior Bernstein estimates on backward time intervals of length comparable to $\min\{t,1\}$ , then give

(7.15)

\sup_{M_{t}^{(j)}}\|\nabla^{m}A(\cdot,t)\|^{2}\leq C\left(1+\frac{1}{t^{m+1}}\right),\qquad t>0.

All constants are independent of the approximating sequence and of the initial curvature. Therefore the estimates give smooth compactness away from $t=0$ and allow to pass to the limit in the smooth approximating flows.

Fix $0<\tau<T<\infty$ . The above estimates give uniform $C^{\infty}$ -bounds for the radial functions $\rho^{(j)}$ on $\mathbb{S}^{n}\times[\tau,T].$ Arzelà–Ascoli’s theorem and a diagonal argument therefore yield a subsequence which converges smoothly on every compact subset of $\mathbb{S}^{n}\times(0,\infty)$ to a smooth positive function

\rho\in C^{\infty}\bigl(\mathbb{S}^{n}\times(0,\infty)\bigr).

Passing to the limit in the radial equation shows that the associated hypersurfaces $M_{t}$ solve (1.3) classically for every $t>0$ .

It remains to identify the initial trace. For the radial parametrization, $\partial_{t}x^{(j)}=\partial_{t}\rho^{(j)}\,\partial_{r}.$ Taking the scalar product with $\nu^{(j)}$ and using (1.3), we obtain

\partial_{t}\rho^{(j)}=\frac{h^{(j)}-H^{(j)}}{\langle\nu^{(j)},\partial_{r}\rangle}.

Estimates (7.6) and (7.14) give

|\partial_{t}\rho^{(j)}|\leq C\left(1+t^{-1/2}\right)\leq Ct^{-1/2},\qquad 0<t\leq 1.

Since the right-hand side is integrable at $t=0$ , integration in time yields

(7.16)

\left\|\rho^{(j)}(\cdot,t)-\rho_{0}^{(j)}\right\|_{L^{\infty}}\leq C\sqrt{t},

where $C$ is independent of $j$ . We know that $\rho_{0}^{(j)}\longrightarrow\rho_{0}\,\text{uniformly on }\mathbb{S}^{n},$ therefore we can pass to the limit in (7.16) and obtain

\|\rho(\cdot,t)-\rho_{0}\|_{L^{\infty}}\leq C\sqrt{t},

which proves ((a)). Since $M_{t}$ and $M_{0}$ are radial graphs over the same sphere, the uniform convergence of their radial functions also implies that $M_{t}\to M_{0}$ in the Hausdorff sense as $t\searrow 0$ .

Each approximating flow is volume-preserving, and starts from initial which, by construction, enclose the same volume as $M_{0}$ . Therefore $\operatorname{vol}(\Omega_{t}^{(j)})=\operatorname{vol}(\Omega_{0}).$ For every fixed $t>0$ , smooth convergence of the radial functions implies convergence of the enclosed volumes. Hence $\operatorname{vol}(\Omega_{t})=\operatorname{vol}(\Omega_{0}).$ In addition, estimates (7.15) pass to the smooth limit on every compact positive-time interval and yield conclusion (b) of Theorem 7.1.

The flow $M_{t}$ we have constructed starts with Lipschitz data and becomes instantaneously smooth. For any $t_{0}>0$ the hypersurface $M_{t_{0}}$ satisfies the hypotheses of the convergence result Theorem 1.1 for smooth initial data, provided the original geometric parameters are chosen sufficiently small. We conclude that the limiting flow becomes strictly convex for all sufficiently large times and converges exponentially in the smooth topology to a round sphere enclosing the same volume as $M_{0}$ .

The construction gives existence through a subsequential limit of smooth approximations. As in [17], no uniqueness assertion is made here for the Lipschitz initial-value problem; in particular, the limiting evolution may a priori depend on the chosen approximating sequence.

We conclude by indicating the modifications required for $K\in\{-1,1\}$ . The proof follows the Euclidean construction and we only record the points at which the ambient geometry changes the argument.

For the approximation, instead of the Euclidean function in (7.2), we use the conformal radial variable

(7.17)

w_{0}=\Psi_{K}(\rho_{0})-\Psi_{K}(R_{0}),\qquad\Psi_{K}^{\prime}(r)=\frac{1}{s_{K}(r)},

for which the condition on the normal in (7.1) again becomes $|\nabla^{\sigma}w_{0}|_{\sigma}\leq\tan\theta_{0}\qquad\text{a.e.}$

The heat-semigroup approximation therefore preserves the same conformal-gradient bound. The enclosed volume is normalized by adding a spatially constant number $b_{j}$ to the approximating conformal radial function. Indeed, since $\Psi_{K}^{\prime}>0$ , the radial function

\rho_{b}=\Psi_{K}^{-1}\bigl(\Psi_{K}(R_{0})+\widetilde{w}_{0}^{(j)}+b\bigr)

is pointwise strictly increasing in $b$ , and hence so is its enclosed volume. There is consequently a unique $b_{j}$ for which the corresponding hypersurface encloses the same volume as $M_{0}$ . Uniform convergence implies $b_{j}\to 0$ , while

\nabla^{\sigma}\bigl(\widetilde{w}_{0}^{(j)}+b_{j}\bigr)=\nabla^{\sigma}\widetilde{w}_{0}^{(j)},

so the volume normalization does not affect the cone condition.

After discarding finitely many terms, Proposition 6.2 applies to all the approximations with a common reflection threshold. The preservation of reflection, Proposition 6.4, and Corollary 6.5 then give constants $0<r_{*}\leq r^{*},\,q_{*}>0,$ independent of the approximation, such that

(7.18)

r_{*}\leq\rho^{(j)}\leq r^{*},\qquad\left\langle\nu^{(j)},\partial_{r}\right\rangle\geq q_{*}.

In the spherical case, the parameters are chosen so that $r^{*}<\frac{\pi}{2}.$ Thus all functions of the radial variable occurring in the evolution equations, including $s_{K}(\rho)$ , $c_{K}(\rho)$ , and the required reciprocals, are uniformly bounded on the range of the approximating flows.

We can next estimate $|A|^{2}$ by the same maximum principle of the Euclidean case, since the terms involving the ambient curvature are of lower order. The estimates on $H$ and on the derivatives of $A$ follow similarly. This allows us to apply the same compactness argument as in the Euclidean case, and to obtain a limiting smooth flow with all the desired properties. This completes the proof of the convergence result for Lipschitz initial data. ∎

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	$\displaystyle\frac{d}{dt}Z_{\max}$	$\displaystyle\leq\frac{\|A\|^{2}}{u-\alpha}(-\alpha Z_{\max}-h)-\frac{h}{u-\alpha}Z_{\max}+2Z_{\max}^{2}$
		$\displaystyle\leq\frac{\|A\|^{2}}{u-\alpha}(-\alpha Z_{\max}-h)-\frac{h}{u-\alpha}\|A\|^{2}+2n\frac{\|A\|^{2}}{(u-\alpha)^{2}}$
		$\displaystyle\leq\frac{\|A\|^{2}}{u-\alpha}\bigg(\frac{2n}{\eta_{0}}-2h-\alpha Z_{\max}\bigg).$

Geometric 𝑪𝟏\bm{C^{1}}-stability of geodesic spheres under volume-preserving mean curvature flow

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification

1. Introduction and main results

Theorem 1.1.

Theorem 1.2 (Nonflat space-form extension).

Theorem 1.3 (Lipschitz initial hypersurfaces).

Relation to previous work and main novelty

Strategy of the proof

Organization of the paper

2. Notation, conventions and preliminary results

Lemma 2.1 (Evolution equations).

Definition 2.2 (Reflection across a hyperplane).

Theorem 2.3 (Preservation of the reflection property; cf. Chow–Gulliver).

3. C1C^{1}-control along the flow via Alexandrov reflection

3.1. Alexandrov reflection property at the initial time

Proposition 3.1.

Proof.

Remark 3.2.

3.2. Reflection and first-order closeness to spheres

Proposition 3.3 (Radial control from reflection).

Proof.

Proposition 3.4 (Normal control from reflection).

Proof.

Corollary 3.5 (Reflection and geometric C1C^{1}-closeness).

Proof.

3.3. Preservation of the starting conditions under the flow

Proposition 3.6 (Annular closeness and normal oscillation for later times).

Proof.

Corollary 3.7 (Control of the support function).

Proof.

Remark 3.8.

4. Curvature control under the flow

4.1. Control of the mean curvature and the global term

Lemma 4.1 (Estimate for the nonlocal term).

Proof.

Remark 4.2 (The sign of the nonlocal term).

Proposition 4.3 (Mean curvature is bounded along the flow).

Proof.

Proposition 4.4 (Shape operator is bounded along the flow).

Proof.

5. Long-time existence and exponential convergence

Proposition 5.1.

Proof.

Proposition 5.2 (Eventual convexity and exponential convergence).

Proof.

6. The result in non-flat space forms

6.1. Geometric C1C^{1}-closeness in space forms

6.2. The initial reflection property

Lemma 6.1 (Reflection of the reference sphere).

Proof.

Proposition 6.2 (Initial reflection in space forms).

Proof.

Lemma 6.3 (Gradient estimate from Alexandrov reflection).

Proof.

Proposition 6.4 (Radial closeness for later times).

Proof.

Corollary 6.5 (Normal control for later times).

Proof.

Corollary 6.6 (Reflection and geometric C1C^{1}-closeness in space forms).

Proof.

6.3. Curvature estimates

Lemma 6.7 (Evolution equations in space forms).

Proposition 6.8 (Uniform mean curvature bound).

Proof.

Proposition 6.9 (Uniform bound for the shape operator).

Proof.

6.4. Subsequential convergence

Proposition 6.10 (Subsequential convergence).

Proof.

6.5. Eventual convexity and exponential convergence

Proposition 6.11 (Eventual convexity and exponential convergence).

Proof.

Corollary 6.12 (Location of the limiting centre).

Proof.

7. Extension to Lipschitz initial data

Theorem 7.1 (Lipschitz stability in space forms).

Proof.

References

Geometric $\bm{C^{1}}$ -stability of geodesic spheres under volume-preserving mean curvature flow

3. $C^{1}$ -control along the flow via Alexandrov reflection

Corollary 3.5 (Reflection and geometric $C^{1}$ -closeness).

6.1. Geometric $C^{1}$ -closeness in space forms

Corollary 6.6 (Reflection and geometric $C^{1}$ -closeness in space forms).