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.     

THE CAUCHY PROBLEMS FOR HIGH DIMENSIONAL QUASILINEAR HYPERBOLIC SYSTEMS

ＴＨＥＣＡＵＣＨＹＰＲＯＢＬＥＭＳＦＯＲＨＩＧＨＤＩＭＥＮＳＩＯＮＡＬＱＵＡＳＩＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＳＹＳＴＥＭＳ￥ＺｈｕＣｈａｎｇｊｉａｎｇ（朱长江）（ＹｏｕｎｇＳｃｉｅｎｔｉｓｔｓＬａｓｏｆａｔｏｒｙｏｆＷｕｈａｎＩｎｓｔ．ｏｆＭａｔ...     

3维双曲空间中给定平均曲率曲面的Weierstrass表示

首先叙述双曲空间中曲面的Ｇａｕｓｓ映照的定义;导出Ｇａｕｓｓ映照所满足的Ｂｅｌｔｒａｍｉ方程;给出了给定平均曲率曲面的Ｗｅｉｅｒｓｔｒａｓｓ表示公式;并讨论了这种表示的完全可积条件．     

3维双曲空间中给定平均曲率曲面的Weierstrass表示

首先叙述双曲空间中曲面的Gauss映照的定义;导出Gauss映照所满足的Beltrami方程;给出了给定平均曲率曲面的Weierstrass表示公式;并讨论了这种表示的完全可积条件.     

NONLINEAR HYPERBOLIC CAUCHY PROBLEMS IN GEVREY CLASSES

91. IntroductionAny linear hyperbolic operator P with Cx coefficients with respect to time, 0 < X < 1, hasa well posed Cauchy problem in suitable Gevrey classes G6 whereas this is not true in Coo.Focusing on operators with characteristics of constaat multip1icity we have the followingbounds for the Gevrey index:1a < he, if P is strictly hyperbolicl','l'5'], (0.1)ta < min{1 X, b}, r 2 2 the largest multip1icity[']. (0.2)Nom (0.1) and (0.2) we have well posedness forra < --3 r 2 1r -- xand…     

双曲空间形式中具有常平均曲率的超曲面(英文)

本文研究了双曲空间形式H~(n+1)(—1)中具有常平均曲率及两个离散主曲率(其中一个主曲率是1-重)的完备连通可定向的n-维超曲面M~n.利用活动标架,得到如果M~n的基本形式的模长满足刚性条件(1.3),那么M~n同构双曲柱面.     

THE SURFACE AREA PRESERVING MEAN CURVATURE FLOW IN QUASI-FUCHSIAN MANIFOLDS

In this paper,we consider the surface area preserving mean curvature flow in quasi-Fuchsian 3-manifolds.We show that the flow exists for all times and converges exponentially to a smooth surface of constant mean curvature with the same surface area as the initial surface.     

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.     

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.     

GLOBAL SMOOTH SOLUTION OF CAUCHY PROBLEMS FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS

In this article we prove that certain kinds of inhomogeneous terms can smoothen thesolution,for first order quasilinear hyperbolic systems globally in time provided that theinitial data are small.     

爱因斯坦流形中超曲面的平均曲率流

本文证明了一个拼嵌的爱因斯坦流形中的任何超曲面在沿其平均曲率向量演化时，如果初发始曲面满足保持其截曲率为正的某些条件，则在有限时间内超曲而将收缩成一点。     

GEOMETRY OF COMPLETE HYPERSURFACES EVOLVED BY MEAN CURVATURE FLOW

Some geometric behaviours of complete solutions to mean curvature flow before the singularities occur are studied. The author obtains the estimates of the rate of the distance between two fixed points and the derivatives of the second fundamental form. By use of a new maximum principle, some geometric properties at infinity are obtained.     

COMPLETE HYPERSURFACES WITH CONSTANT MEAN CURVATURE AND FINITE INDEX IN HYPERBOLIC SPACES

In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4(-1)(H5(-1)) with constant mean curvature H satisfying H2 > 6634 (H2 > 114785 respectively) must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4(-1) (resp. H5(-1)) with constant mean curvature H satisfying H2 > 6643 (resp. H2 > 114785 ) must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].     

ERROR ESTIMATES FOR THE FULL-DISCRETE MIXED FEM FOR NONLINEAR HYPERBOLIC PROBLEMS

1 IntroductionConsider the second order nonlinear hyPErbolic equationwhere D be a bounded domain in R' with Lipschitz boundary are, and J~ (0,TJ.The following regularity assumptions will be made on the functions a,c,f,g and solutionu of (1. I ):(1) there exist constantS c.,c*,a., and a' such that for all xos and ie R,(2) The functions a ~a (x, u),c ~ c(x, u ), f~f(x, u,t), g~g (x, t) are continuously differentiable with respect to u and t. Moreover, there exists a bound K= such that, for …     

某类一阶非线性双曲型方程的Cauchy问题

本文考虑一类具实际物理背景的一阶非线性双曲型方程，证明了解的存在唯一及正则性。     

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL

We obtain the Omori-Yau maximum principle on complete properly immersed submanifolds with the mean curvature satisfying certain condition in complete Riemannian manifolds whose radial sectional curvature satisfies some decay condition, which generalizes our previous results in [17]. Using this generalized maximum principle, we give an estimate on the mean curvature of properly immersed submanifolds in H^n × R^e with the image under the projection on H^n contained in a horoball and the corresponding situation in hyperbolic space. We also give other applications of the generalized maximum principle.     

中曲率大于零的非凸曲面

本文研究了曲面的曲率问题.利用积分几何方法和旋转曲面性质,构造出一个欧氏空间R3中紧致光滑的中曲率H大于零的非凸曲面,并得到了关于紧致光滑曲面曲率的几个不等式.     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.