HYPERBOLIC MEAN CURVATURE FLOW:EVOLUTION OF PLANE CURVES

In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F ： S1 × [0, T ） → R^2 which satisfies the following evolution equation δ^2F /δt^2 （u, t） = k（u, t）N（u, t）-▽ρ（u, t）, ∨（u, t） ∈ S^1 × [0, T ） with the initial data F （u, 0） = F0（u） and δF/δt （u, 0） = f（u）N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F （u, t）, f（u） and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=（δ^2F /δsδt ,δF/δt） T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax） and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.

SELF-SIMILAR SOLUTIONS TO THE HYPERBOLIC MEAN CURVATURE FLOW

This article concerns the self-similar solutions to the hyperbolic mean curvature flow （HMCF） for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.

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. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.

SOME EXTENSIONS OF THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case

Symmetries and conservation laws associated with a hyperbolic mean curvature flow

Under investigation in this paper is a hyperbolic mean curvature flow for convex evolving curves.Firstly,in view of Lie group analysis,infinitesimal generators,symmetry groups and an optimal system of symmetries of the considered hyperbolic mean curvature flow are presented.At the same time,some group invariant solutions are computed through reduced equations.In particular,we construct explicit solutions by applying the power series method.Furthermore,the convergence of the solutions of power series is certificated.Finally,conservation laws of the hyperbolic mean curvature flow are established via Ibragimov's approach.

SYMMETRY OF TRANSLATING SOLUTIONS TO MEAN CURVATURE FLOWS

First, we review the authors＇ recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.

Mean Curvature Flow of Arbitrary Codimension in Complex Projective Spaces

Recently,Pipoli and Sinestrari[Pipoli,G.and Sinestrari,C.,Mean curvature flow of pinched submanifolds of CPn,Comm.Anal.Geom.,25,2017,799-846]initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CPm.The purpose of this paper is to develop the work due to Pipoli and Sinestrari,and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space.Namely,the authors prove that if the initial submanifold in CPm satisfies a suitable pinching condition,then the mean curvature flow converges to a round point in finite time,or converges to a totally geodesic submanifold as t→∞.Consequently,they obtain a differentiable sphere theorem for submanifolds in the complex projective space.

Ginzburg-Landau Vortex and Mean Curvature Flow with External Force Field

This paper is devoted to the study of the vortex dynamics of the Cauchy problem for a parabolic Ginzburg Landau system which simulates inhomogeneous type II superconducting materials and three-dimensional superconducting thin films having variable thickness. We will prove that the vortex of the problem is moved by a codimension k mean curvature flow with external force field. Besides, we will show that the mean curvature flow depends strongly on the external force, having completely different phenomena from the usual mean curvature flow.

Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface E is a graph in R^4 if there is a unit constant 2-form ω on R^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 v_0≥on the initial snrface,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.ω>≥v_0>0 where w_1 is a Khler form on M_1.We prove that.if M is a Khler-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.

Translating Surfaces of the Non-parametric Mean Curvature Flow in Lorentz Manifold M^(2)×R^(*)

In this paper, for the Lorentz manifold M^(2)× R with M^(2) a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M^(2), which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.

Convex Mean Curvature Flow with a Forcing Term in Direction of the Position Vector

A smooth, compact and strictly convex hypersurface evolving in R^n＋1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.

ε_0-Regularity for Mean Curvature Flow from Surface to Flat Riemannian Manifold

In this paper we prove an so-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫∑0 ｜A｜^2 ≤ ε0 and the initial area u0（∑0） is not large, then along the mean curvature flow, we have ∫∑t ｜A｜^2 ≤ ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold M^(n)×R^(*)

In this paper,the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold M^(n)×R,where M^(n) is an n-dimensional(n≥2)complete Riemannian manifold with nonnegative Ricci curvature,and R is the Euclidean 1-space.

Rotationally Symmetric Translating Solutions to Curvature Flows in Image Processing

This paper proves the existence of rotationally symmetric solutions to a curvature flow in image processing. The flow includes the level sets flow and the mean curvature flow projected onto the normal. Sharp estimates are obtained for these solutions..

Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3

By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C^(2)-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.

Mean curvature flow with linear oblique derivative boundary conditions

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.Also,we prove the asymptotic behavior of the mean curvature flow with zero oblique derivative boundary data which is a generalization of Huisken’s original result about prescribed perpendicular contact angle.

Singularities of mean curvature flow

Mean curvature flow and its singularities have been paid attention extensively in recent years. The present article reviews briefly their certain aspects in the author's interests.

Anisotropic inverse harmonic mean curvature flow

We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^＊ 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^＊, then H（.,t） remains uniformly convex and expands to infinity as t →∞ and its scaling, H（-, t）e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.

Inverse Curvature Flows of Rotation Hypersurfaces

We consider the inverse curvature flows of smooth,closed and strictly convex rotation hypersurfaces in space forms M_(κ)^(n+1)with speed function given by F^(-α),whereα∈(0,1]forκ=0,-1,α=1 forκ=1 and F is a smooth,symmetric,strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces.We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value,and prove the long time existence and convergence of the flows.No second derivatives conditions are required on F.

A Note on Harnack Type Inequality for the Gaussian Curvature Flow

In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow’s Harnack inequality and entropy estimate.