等距曲面的分析性质

研究了等距曲面的局部分析性质 ,给出了等距曲面第一、二基本形式 ,平均曲率 ,高斯曲率的具体表达式 .     

高维常平均曲率超曲面的数量曲率的空隙

本文证明了当高维球面中闭常平均曲率超曲面Ｍ＾ｎ的平均曲率│Ｈ│＜Ｃ（ｎ）时，若Ｍ的数量曲率Ｒ为常数，则Ｒ不属于［ａ（ｎ，Ｈ），ａ（ｎ，Ｈ）＋ｎ／４），其中ａ（ｎ，Ｈ）＝ｎ（ｎ－９／４）＋ｎ＾２（ｎ－２）／２（ｎ－１）Ｈ＾２－ｎ（ｎ－２）／２（ｎ－１）√ｎ＾２Ｈ＾４＋４（ｎ－１）Ｈ＾２。     

空间形式中具有平均曲率的曲面上强稳定域曲率估计

The purpose of this paer is to establish some identities from generalized Legendre polynoials.These results eneralize the results of P.C.Joshi and N.Balakrishnon in [1].     

三维欧氏空间的仿射乘积曲面

定义了三维欧氏空间中的仿射乘积曲面,给出了极小仿射乘积曲面以及高斯曲率为零的仿射乘积曲面的分类,同时还给出了平均曲率为非零常数的仿射乘积曲面的分类.     

常数平均曲率子流形的曲率估计及其应用

本和用活动标架法及Laplacian特征值方法研究了空间形中具有常数平均曲率的子流形，给出了高斯曲率与数量曲率的一种估计方法，证明了空间形中具有常数平均曲率的子流形上一个单连通有界区域为稳定的两个充分条件。     

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.     

三维空间形式中常平均曲率曲面的稳定性

本文利用共形度量高斯曲率的估计研究了三维空间形式N^3(C）中具常平均曲率曲面的区域稳定性。     

H~2×R中给定平均曲率曲面的Weierstrass表示

定义了H~2×R中曲面的法高斯映照,给出了给定平均曲率曲面的Weierstrass表示,证明了法高斯映照满足一个二阶偏微分方程,并且该方程是所得到的Weierstrass表示的完全可积条件.     

R~3中正曲率曲面的无穷小变形和几类边值问题

本文讨论了Ｒ３中正曲率曲面的无穷小变形，利用广义解析函数理论，研究了几类有明确几何意义的边值问题的无穷小刚性，其中部分结论是Ｖｅｋｕａ相应结果的推广．     

H~2×R中给定平均曲率曲面的Weierstrass表示

定义了H~2×R中曲面的法高斯映照,给出了给定平均曲率曲面的Weierstrass表示,证明了法高斯映照满足一个二阶偏微分方程,并且该方程是所得到的Weierstrass表示的完全可积条件.     

三维欧氏空间中的一类Weingarten螺旋面

通过求解相关的非线性常微分方程,构造了三维欧氏空间中主曲率之差为常数的螺旋面,并证明这类曲面的广泛存在性.     

Indefinite Affine Umbilical Surfaces in R^4

We study surfaces M ² in the four-dimensional affine space equipped with its usual torsion-free connection D and parallel volume form given by the determinant.     

R^4中的伪脐曲面及Gauss映射

本文研究了伪脐曲面M的一些性质,基于它并利用M的Ganuss映射,给出了M作为R^4中平坦环面的一个充分条件。     

A Theorem on Infinitesimal I-isometry of Surfaces Immersed in a Space with Constant Curvature

1.IntroductionTheproblemsofinfinitesimalrigidityandisometricdeformationhavebeenextensivelyin-vestigatedforsurfacesinthreedimensionalEuclideanspaceE'.See,forexample,A.SvecL1]'E.Kann[2j,ZhouJiazu[3Jandthepapers[4j,[5J,[6JbyYangWenmao3also,KobayashiandNomizu[7].ThefirstauthorandZhangGaoyonggeneralizedthetheoryofinfinitesimali-sometriesofsurfacesinE3tothehypersurfacesimmersedinaspaceofconstantcurvature[8j.K.Tenenblat[9]andR.A.GoldsteinandP.J.Ryan[1o]studiedtheinfinitesimalI-isometryfors…     

Surfaces with harmonic inverse mean curvature in space forms

We define surfaces with harmonic inverse mean curvature in space forms and generalize a theorem due to Lawson by which surfaces of constant mean curvature in one space form isometrically correspond to those in another. We also obtain an immersion formula, which gives a deformation family for these surfaces.

     

Evolution equation of the Gauss curvature under hypersurface flows and its applications

In this paper, we derive evolution equation of the integral of the Gauss curvature on an evolving hypersurface. As an application, we obtain a monotone quantity on the level surface of the potential function on a 3-dimensional steady gradient Ricci soliton with positive sectional curvature, and prove that such a soliton is rotationally symmetric outside of a compact set under a curvature decaying assumption. Along the way we will also apply our evolution equation to some other cases.     

参数曲面的局部凸性条件

参数曲面的凸性分析在计算机辅助设计中有着重要的作用.给出了一般参数曲面局部凸的定义,利用曲面的第二基本量和高斯曲率.得到了一般参数曲面局部凸的几个必要条件.     

微分几何观点下二元函数的极值

利用曲面的局部微分性质给出二元函数极值存在的必要条件和充分条件,并将之运用于具有明显几何特征的曲面对应的二元函数极值的判别问题中.     

单位球中的极小曲面

Let M be a closed surface with positive Gauss curvature minimally immersed in a standard Euclidean unit sphere S~n.In this paper,we choose a local orthonormal frame field on M,under which the shape operators have very convenient form.We also give some applications of this kind of frame field.     

A Note on Doubly Connected Surfaces of Constant Mean Curvature with Prescribed Boundary

In this paper we prove the existence of a constant mean curvature surface spanning two given convex curves in parallel planes of ℝ³ under hypotheses relating the distance between the planes, the curvature of the curves and the mean curvature. It is also proved that the surface is a radial graph over a unit sphere.This author was partially Supported by Fapergs. Mathematics Subject Classifications (2000): 53A10, 53C42, 35J60