On the Conditions of Extending Mean Curvature Flow

The authors consider a family of smooth immersions F(·,t):Mn → Rn+1 of closed hypersurfaces in Rn+1 moving by the mean curvature flow F∈(pt,t)=-H(p,t)·ν(p,t) for t ∈ [0,T).They show that if the norm of the second fundamental form is bounded above by some power of mean curvature and the certain subcritical quantities concerning the mean curvature integral are bounded,then the flow can extend past time T.The result is similar to that in [6-9].     

Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ^{n + 1} remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1 , and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.     

Long-Time Behavior of the Mean Curvature Flow with Periodic Forcing

We consider the long-time behavior of the mean curvature flow in heterogeneous media with periodic fibrations, modeled as an additive driving force. Under appropriate assumptions on the forcing term, we show existence of generalized traveling waves with maximal speed of propagation, and we prove the convergence of solutions to the forced mean curvature flow to these generalized waves.     

The Second Type Singularities of Symplectic and Lagrangian Mean Curvature Flows

This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H|² on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.     

Generalized Mean Curvature Flow in Carnot Groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4 Chen , Y. , Giga , Y. , Goto , S. ( 1991 ). Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations . J. Diff. Geom. 33 : 749 – 786 .[Crossref], [Web of Science ®] , [Google Scholar]] and [12 Evans , L. C. , Spruck , J. ( 1991 ). Motion of level sets by mean curvature. I . J. Diff. Geom. 33 ( 3 ): 635 – 681 . [Google Scholar]]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

The First Eigenvalue of p-Laplace Operator Under Powers of the mth Mean Curvature Flow

In this paper, we derive the evolution equation for the eigenvalues of p-Laplace operator. Moreover, we show the following main results. Let ( ${M^{n}, g(t)), t\in [0,T),}$ be a solution of the unnormalized powers of the mth mean curvature flow on a closed manifold and λ_1,p (t) be the first eigenvalue of the p-Laplace operator (p ≥ n). At the initial time t = 0, if H > 0, and $$h_{ij}\geq \varepsilon Hg_{ij}\quad \left(\frac{1}{p}\leq\epsilon\leq \frac{1}{n}\right),$$ then λ_1,p (t) is nondecreasing and differentiable almost everywhere along the unnormalized powers of the mth mean curvature flow on [0,T). At last, we discuss some interesting monotonic quantities under unnormalized powers of the mth mean curvature flow.     

Mean Curvature Flow with Convex Gauss Image

In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.     

Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

The author establishes the long-time existence and convergence results of the mean curvature flow of entire Lagrangian graphs in the pseudo-Euclidean space, which is related to the logarithmic Monge-Ampère flow.     

Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface Σ is a graph in ?⁴ if there is a unit constant 2-form ω on ?⁴ such that <e ₁∧e ₂, ω≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if $ \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} A surface Σ is a graph in ℝ⁴ if there is a unit constant 2-form ω on ℝ⁴ such that <e ₁∧e ₂, ω>≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if v ₀≥ on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M ₁×M ₂ where M ₁ and M ₂ are Riemann surfaces, if <e ₁∧e ₂, ω₁>≥v ₀>0 where ω₁ is a K?hler form on M ₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, v ₀≥ on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic. Received July 25, 2001, Accepted October 11, 2001     

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].     

Mean Curvature Flow via Convex Functions on Grassmannian Manifolds

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the v-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the v-function. Under such restrictions, curvature estimates in terms of v-function composed with the Gauss map can be carried out.     

A Relation between Mean Curvature Flow Solitons and Minimal Submanifolds

We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space where equals ℝ × N equipped with a warped product metric and show that a submanifold inN converges to a conformal soliton under the mean curvature flow in N if and only if its associatedsubmanifold in converges to a minimal submanifold under a rescaled mean curvature flow in . We then define a notion of stability for conformal solitons and obtain L^p estimates as well as pointwise estimates for the curvature of stable solitons.     

The Inverse Mean Curvature Flow in Rotationally Symmetric Spaces

In this paper, the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied. It is proved that the flow converges to a unique geodesic sphere, i.e., every principle curvature of the hypersurfaces converges to a same constant under the flow.     

The Extension of the H~k Mean Curvature Flow in Riemannian Manifolds

In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").     

Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation

We discuss the motion of noncompact axisymmetric hypersurfaces Γ _t evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with the shrinking cylinder evolved by the flow and prove that they tend to a smooth hypersurface having no pinching neck and having closed ends at infinity of the axis of rotation as the quenching time is approached. Moreover, they are completely characterized by a condition on initial hypersurface.     

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.     

Asymptotic Behavior of Sample Mean Direction for Spheres

In this note we consider some asymptotic properties of empirical mean direction on spheres. We do not require any symmetry for the underlying density. Thus our results provide the framework for an asymptotic inference regarding mean direction under very weak model assumptions. Mean direction is a specialization of the more general concept of mean location applicable to arbitrary (compact) submanifolds of Euclidean space, to which the methods of this paper could be applied.