The mean curvature flow in Minkowski spaces

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we prove the short time existence and uniqueness for solutions of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.     

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface M _t meets such tgh orthogonally, we prove that: (a) the flow exists while M _t does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M _t remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set (if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.     

The Ricci flow on the two ball with a rotationally symmetric metric

In this paper we study a boundary-value problem for the Ricci flow in the two-dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. The text was submitted by the author in English.     

常均曲率热流的存在性

给出常均曲率热流的Dirichlet边值问题存在唯一和正则的解,并且这个解可以一直达到某个能量集中的时刻.如果这个解还满足一定的能量不等式,那么可以得到在除有限个奇点的全局解.我们所使用的方法有别于文献[2].     

Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension

We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface.     

Mean curvature flow with a forcing term in minkowski space

We study the forced mean curvature flow of graphs in Minkowski space and prove longtime existence of solutions. When the forcing term is a constant, we prove convergence to either a constant mean curvature hypersurface or a translating soliton – depending on the boundary conditions at infinity. It is a pleasure to thank my PhD advisors Klaus Ecker and Gerhard Huisken for their assistance and encouragement. I also thank Maria Athanassenas, Oliver Schnürrer and Marty Ross for their interest and useful comments, and the Max Planck Gesellschaft for financial support.     

Non-parametric radially symmetric mean curvature flow with a free boundary

We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.     

Existence of constant mean curvature graphs in hyperbolic space

We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible. Received July 18, 1997 / Accepted April 24, 1998     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.     

Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R~(n+1)between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.     

Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition

In this work we study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface. We deduce a very sharp local gradient bound depending only on the curvature of the immersions and. Combining this with a short time existence result, we obtain the existence of a unique solution to any given smooth initial and boundary data. This solution either exists for anyt>0 or on a maximal finite time interval [0,T] such that the curvature explodes astT.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Mean curvature flow with linear oblique derivative boundary conditions

Science China Mathematics - In this paper, we study the mean curvature flow with oblique derivative boundary conditions. We prove the longtime existence by choosing a suitable auxiliary function....     

Sufficient conditions for constant mean curvature surfaces to be round

We study under what condition a constant mean curvature surface can be round: i) If the boundary of a compact immersed disk type constant mean curvature surface in consists of lines of curvature and has less than 4 vertices with angle , then the surface is spherical; ii) A compact immersed disk type capillary surface with less than 4 vertices in a domain of bounded by spheres or planes is spherical; iii) The mean curvature vector of a compact embedded capillary hypersurface of with smooth boundary in an unbounded polyhedral domain with unbalanced boundary should point inward; iv) If the kth order () mean curvature of a compact immersed constant mean curvature hypersurface of without boundary is constant, then the hypersurface is a sphere. Received: 3 October 2000 / Published online: 1 February 2002     

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann–Dirichlet boundary conditions and more general curvature-dependent speeds.     

Axially symmetric volume constrained anisotropic mean curvature flow

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume in comparison with their initial surface energy, we show that the flow exists for all time. Numerical simulations of the curvature flow are presented.     

Mean Curvature Flow with Bounded Gauss Image

We study the mean curvature flow of a complete space-like submanifold in pseudo-Euclidean space with bounded Gauss image and bounded curvature. We establish a relevant maximum principle for our setting. Then, we can obtain the ??confinable property?? of the Gauss images and curvature estimates under the mean curvature flow. Thus we prove a corresponding long time existence result.     

Nonparametric Mean Curvature Flow with Nearly Vertical Contact Angle Condition

For any bounded strictly convex domain $\Omega$ in $\mathbb{R}^n$ with smooth boundary, we find the prescribed contact angle which is nearly perpendicular such that nonparametric mean curvature flow with contact angle boundary condition converge to ones which move by translation. Subsequently, the existence and uniqueness of smooth solutions to the capillary problem without gravity on strictly convex domain are also discussed.