If $ψ:M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional $\mathcal{F}_m(ψ) = \int_M 1 + |\nabla^m ν|^2 \, dμ$, where $ν$ is a local unit normal vector along $ψ$, $\nabla$ is the Levi-Civita connection of the Riemannian manifold $(M,g)$, with $g$ the pull-back metric induced by the immersion and $μ$ the associated volume measure. We prove that if $m>\lfloor n/2 \rfloor$ then the unique globally defined smooth solution to the $L^2$-gradient flow of $\mathcal{F}_m$, for every initial hypersurface, smoothly converges asymptotically to a critical point of $\mathcal{F}_m$, up to diffeomorphisms. The proof is based on the application of a Lojasiewicz-Simon gradient inequality for the functional $\mathcal{F}_m$.
We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0<p\leq 1.$ If $f$ is the mean curvature, we obtain stronger Harnack inequalities.
We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the background manifold is sufficiently pinched and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.
In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second fundamental form of these hypersurfaces in some space-time region $B_R(y)\times (0, t_0)$ implies that the $\mathcal{H}^{n+2-p}$ - measure of the first singular set vanishes inside $B_R(y)$. For $p=2$, this was established independently by Han and Sun. Our result furthermore generalizes previous work of Xu, Ye and Zhao and of Le and Sesum for $p\ge n+2$, in which case…
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed using level-set methods and an appropriate variational principle. This new flow has a natural application as a variational-type approach to constructing MOTS, and this work also gives new insights into the theory of weak solutions of inverse mean curvature flow.
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a class of inverse curvature flows, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.
We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hypersurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the pri…
We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface.
A recent article Li and Lv considered contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in certain cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we extend the result to various other cases that are analogous to those considered in other earlier work, and we show that in all cases, where sufficient pinching conditions are assumed on the initial hypersurface, then under suitable rescaling the final point is asymptoti…
We consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface. We study this problem by level set method. We give some criteria to judge whether the interface evolution is fattening or non-fattening.
We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide several rigidity results under appro- priate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi-Bernstein type results are given.
We address the asymptotic behavior of the $α$-Gauss curvature flow, for $α>1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface.
Huisken and Sinestrari have recently defined a surgery process for mean curvature flow when the initial data is a two-convex hypersurface. The process depends on a parameter H. Its role is to initiate a surgery when the maximum of the mean curvature of the evolving hypersurface becomes H, and to control the scale at which each surgery is performed. We prove that as H goes to infinity the surgery process converges to level set flow.
We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed normal derivative satisfy a uniform oscillation estimate.
For a mean curvature flow of complete graphical hypersurfaces $M_{t}=\operatorname{graph} u(\cdot,t)$ defined over domains $Ω_{t}$, the enveloping cylinder is $\partialΩ_{t}\times\mathbb{R}$. We prove the smooth convergence of $M_{t}-h\,e_{n+1}$ to the enveloping cylinder under certain circumstances. Moreover, we give examples demonstrating that there is no uniform curvature bound in terms of the inital curvature and the geometry of $Ω_{t}$. Furthermore, we provide an example where the hypersurface increasingly oscillates towards infinity in both space and time. It has unbounded curvature at a…
We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.
We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. In dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.
We investigate the evolution of closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$, n=3, for contracting normal velocities, including powers of the mean curvature, of the norm of the second fundamental form, and of the Gauss curvature. We prove convergence to a round point for 2-pinched initial hypersurfaces. In $\mathbb{R}^{n+1}$, n=2, natural quantities exist for proving convergence to a round point for many normal velocities. Here we present their counterparts for arbitrary dimensions $n\in\mathbb{N}$.
This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_κ^{n+1}$ of constant sectional curvature $κ$ evolving in direction of its normal vector, where the speed is given by a power $β(\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gauß}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $β$ and $κ$, then under t…
We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.
We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem under the weakly pinching condition. In particular, we obtain a classification theorem for weakly pinched hypersurfaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial hypersurface is positive, but does not imply positivity of the…
We study the rescaled mean curvature flow (MCF) of hypersurfaces that are global graphs over a fixed cylinder of arbitrary dimensions. We construct an explicit stable manifold for the rescaled MCF of finite codimensions in a suitable configuration space. For any initial hypersurface from this stable manifold, we construct a unique global solution to the rescaled MCF, and derive precise asymptotics for these solutions that are valid for all time. Using these asymptotics, we prove asymptotic stability of cylindrical singularities of arbitrary dimensions under generic initial perturbations. As a …
Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.
We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when the hypersurface has a free boundary and satisfies Neumann boundary data, and to a sphere when it is compact without boundary.
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which actually cor…
In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.
The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we obtain, after normalization, a symmetric bi-linear form Q. In this paper, we prove that Q converges to the affine Blaschke metric when we approximate the hypersurface along a curve whose points are centroids of parallel sections. We also show that the rate of this convergence is gi…
In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^ασ_2^{1/2},$ where $σ_2^{1/2}$ is the normalized square root of the scalar curvature, $α\geq 2,$ and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the starshapedness is preserved. Moreover, after normalization, we show that the flow converges exponentially fast to a sphere centered at origin. When $α<2,$ a counterexample is given for the above convergence.
In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^{\fracαβ}σ_k^{\frac{1}β}$, where $σ_k$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $α$, $β$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $α$, $β$. When $k\geq2$, $0<β\leq 1$, $α\geq β+k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after …
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which a…
In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of other lower order anisotropic curvature measures with nonnegative coefficients is a scaled Wulff shape. This generalizes the classical results by Schneider [Comment. Math. Helv. \textbf{54} (1979), 42--60] and by Kohlmann [Arch. Math. (Basel) \textbf{70} (1998), 250--256] to the anisotropic setting. The main ingredients in the proof are the generalized anis…
We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH \equiv 0) hypersurface in \R^3 or \R^4 with restricted growth of the curvature at infinity and small total tracefree curvature must be an embedded union of umbilic hypersurfaces. Then we prove for surfaces that if the L^2 norm of the tracefree curvature is globally initially small it is monotonic nonincreasing along the flow. We also derive pointwise estimates for all derivatives of the curvatur…
In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with the normal speed $Φ$ given by $r^αF^β$ or $u^αF^β$, where $F$ is a monotone, symmetric, inverse-concave, homogeneous of degree one function of the principal curvatures, $r$ is the distance from the hypersurface to the origin and $u$ is the support function of hypersurface. If $α\geq β+1$ when $Φ=r^αF^β$ or $α> β+1$ when $Φ=u^αF^β$, we prove that the flow exists for all times and converges to the origin. After proper rescaling, we…
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piec…
We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Saez and the second author yields a global weak solution to the original problem for general initial data and onesided obstacles.
In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\mathbb{R}^{n+1}$ and $\mathbb{S}^{n+1}$ by $σ_k^α$, where $σ_k$ is the $k$-th elementary symmetric function of the principal curvatures and $α\ge 1/k$. We prove that for any $n\geq3$ and any fixed $k$ with $1\leq k\leq n$, there exists a constant $c(n,k)>1/k$ such that that if $α$ lies in the interval $[1/k,c(n,k)]$, then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hyper…
We study the flow $M_t$ of a smooth, strictly convex hypersurface by its mean curvature in $\mathrm{R}^{n+1}$. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time $T$ and point $x^*$ (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere ${\bf S^n}$ of radius $\sqrt{n}$. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us the rate of the exponential decay is at least $\frac{2}{n}$. We can define …
We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows on torus fibered minimal models to obtain convergence results.
