Mean Convex Hypersurfaces Research Articles

A Heintze-Karcher inequality with free boundaries and applications to capillarity theory

In this paper we analyze the shape of a droplet inside a smooth container. To characterize their shape in the capillarity regime, we obtain a new form of the Heintze–Karcher inequality for mean convex hypersurfaces with boundary lying on curved substrates.

Nonhomogeneous expanding flows in hyperbolic spaces

In the present paper, we consider star-shaped mean convex hypersurfaces of the real, complex and quaternionic hyperbolic space evolving by a class of nonhomogeneous expanding flows. For any choice of the ambient manifold, the initial conditions are preserved and the long-time existence of the flow is proved. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling, the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space; otherwise, it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every case, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

Locally constrained inverse mean curvature flow in GRW spacetimes

<p style='text-indent:20px;'>We obtain the long-time existence and smooth convergence results for the locally constrained inverse mean curvature flow of compact spacelike hypersurface in certain GRW spacetimes. As an application of the convergence result, we establish a weighted Minkowski type inequality for closed spacelike mean-convex hypersurface in de Sitter space under mild rigidity assumption.</p>

Low entropy and the mean curvature flow with surgery

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $$\varLambda _{n-2}$$ . In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $$H - \frac{\langle x, \nu \rangle }{2} > 0$$ is possible and as an application we use the surgery flow to show that smooth n-dimensional closed self shrinkers with entropy less than $$\varLambda _{n-2}$$ are isotopic to the round n-sphere.

Weighted geometric inequalities for hypersurfaces in sub‐static manifolds

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curvature and the area of the hypersurface. Versions of the first inequality for the sphere and for the adS-Reissner-Nordstr\"om manifold are proven. We end with an example of a convex surface for which the ratio between the polar moment of inertia and the square of the area is less than that of the round sphere.

Inverse curvature flow for hypersurfaces in Riemannian manifold and its application

This is a survey paper on the inverse curvature flow for hypersurfaces in Riemannian manifold.We first discuss the long time behavior of the inverse curvature flow in Euclidean space, and its application in proving the Alexandrov-Fenchel inequalities for star-shaped hypersurfaces.Then we discuss the related results in hyperbolic space and in sphere.Finally, we discuss the inverse mean curvature flow in Kottler space. Kottler space is an important example of warped product space,and is aymptotically locally hyperbolic at the infinity and satisfies the static equation. We will consider the convergence result of inverse mean curvature flow in such space and also discussits application in proving the Minkowski-type inequality for star-shaped and mean convex hypersurfaces. Inverse curvature flow is an active research area in recent years.We cannot include all results in this short article. For the convenience of the interested readers, we list a few related references on other topics that we do not mention.

Willmore inequality on hypersurfaces in hyperbolic space

In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic $3$-space.

A survey on Inverse mean curvature flow in ROSSes

Abstract In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.

Inverse Mean Curvature Flows in Warped Product Manifolds

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of $\varphi''(r)$ and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.

Brendle’s Inequality on Static Manifolds

We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chru\'{s}ciel and Simon \cite{CS}.

An inscribed radius estimate for mean curvature flow in Riemannian manifolds

We consider a family of embedded, mean convex hypersurfaces in a Riemannian manifold which evolve by the mean curvature flow. We show that, given any number $T>0$ and any $\delta>0$, we can find a constant $C_0$ with the following property: if $t \in [0,T)$ and $p$ is a point on $M_t$ where the curvature is greater than $C_0$, then the inscribed radius is at least $\frac{1}{(1+\delta) \, H}$ at the point $p$. The constant $C_0$ depends only on $\delta$, $T$, and the initial data.

Mean Curvature Flow of Mean Convex Hypersurfaces

In the last 15 years, White and Huisken‐Sinestrari developed a far‐reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high‐curvature regions in a mean convex flow.In the present paper, we give a new treatment of the theory of mean convex (and k‐convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem.Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new noncollapsing result of Andrews [2]. Some parts are also inspired by the work of Perelman [32,33]. In a forthcoming paper [17], we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper.Note added in May 2015. Since the first version of this paper was posted on arxiv in April 2013, the estimates have been used to construct mean convex flow with surgery in ℝ3 by Brendle and Huisken [5] in September 2013 and in another paper by the authors in April 2014.© 2016 Wiley Periodicals, Inc.

Subsequent singularities of mean convex mean curvature flows in smooth manifolds

For any n-dimensional smooth manifold \(\Sigma \), we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in \(\Sigma \) are cylindrical (of convex type) if the flow converges to a smooth hypersurface \(M_\infty \) (maybe empty) at infinity. Previously this was shown (1) for \(n\le 7\), and (2) for arbitrary n up to the first singular time without the smooth condition on \(M_\infty \).

An Example of a Mean-Convex Mean Curvature Flow Developing Infinitely Many Singular Epochs

In this paper, we give an example of a compact mean-convex hypersurface with a single singular point moved by mean curvature having a sequence of singular epochs (times) converging to zero.

An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality

We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This sharpens previous results by Dahl-Gicquaud-Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space-times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chrusciel and Simon on the validity of a Penrose-type inequality for exotic black holes.

A note on the entropy of mean curvature flow

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We will also prove that a smooth self-shrinker with low entropy is exact a hyperplane.

Subsequent singularities in mean-convex mean curvature flow

We use Ilmanen’s elliptic regularization to prove that for an initially smooth mean convex hypersurface in \(\mathbf {R}^n\) moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. In particular, the tangent flows are all shrinking spheres or cylinders. Previously this was known only (1) for \(n\le 7\), and (2) for arbitrary \(n\) up to the first singular time.

A sharp bound for the inscribed radius under mean curvature flow

We consider a family of embedded, mean convex hypersurfaces which evolve by the mean curvature flow. It follows from general results of White that the inscribed radius at each point on the hypersurface is at least $$\frac{c}{H}$$ , where $$c$$ is a positive constant that depends only on the initial data. Andrews recently gave a new proof of that fact using the maximum principle. In this paper, we show that the inscribed radius is at least $$\frac{1}{(1+\delta ) \, H}$$ at each point where the curvature is sufficiently large.

On Brendle’s Estimate for the Inscribed Radius Under Mean Curvature Flow

In a recent paper [3], Brendle proved that the inscribed radius of closed embedded mean convex hypersurfaces moving by mean curvature flow is at least |$\frac {1}{(1+\delta)H}$| at all points with H≥C(δ,M0). In this note, we give a shorter proof of Brendle’s estimate, and of a more general result for α-Andrews flows, based on our recent estimates from Haslhofer and Kleiner [4].

A note on the mean curvature flow in Riemannian manifolds

Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G. Huisken and C. Sinestrari in [5]. These a-priori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.