关于Weingarten超曲面的一个存在性定理

本文研究伪黎曼空间型中一类特殊的Weingarten超曲面的存在性问题.通过引入伪黎曼空间型中旋转超曲面的概念,并给出其主曲率计算公式.得到伪黎曼空间型中旋转型Weingarten超曲面的存在性定理.推广并统一了相关结果.     

R2,1中类时Weingarten曲面的Bcklund变换

本文利用平行变换,得到3维Minkowski空间R2,1中的K-2mH+m2-l2=0(H2>K,0≤m相似文献   

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^P-norm Curvature in R^n＋1

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.     

Weingarten曲面和Sine-Gordon方程

建立了满足条件K-2mH+m²+l²=0的Weingarten曲面和Sine-Gordon方程的解之间的联系。     

Hypersurfaces with Constant Mean Curvature in Space Forms

Ｈｙｐｅｒｓｕｒｆａｃｅｓ　ｗｉｔｈ　Ｃｏｎｓｔａｎｔ　Ｍｅａｎ　Ｃｕｒｖａｔｕｒｅ　ｉｎ　Ｓｐａｃｅ　ＦｏｒｍｓＨｙｐｅｒｓｕｒｆａｃｅｓｗｉｔｈＣｏｎｓｔａｎｔＭｅａｎＣｕｒｖａｔｕｒｅｉｎＳｐａｃｅＦｏｒｍｓ￥ＳｏｎｇＨｏｎｇｚａｏ；ＨｕＺｅｊ...     

R2,1中类时Weingarten曲面的B?cklund变换

本文利用平行变换，得到３维Ｍｉｎｋｏｗｓｋｉ空间Ｒ^2,1中的Ｋ－２ｍＨ＋ｍ²－ｌ²＝０（Ｈ²＞ Ｋ，０≤ｍ＜ｌ）类时Ｗｅｉｎｇａｒｔｅｎ曲面的Ｂ?ｃｋｌｕｎｄ定理．     

R2,1中线性Weingarten曲面的B(a)cklund变换

本文对三维Minkowski空间R2,1中具有性质K-2mH+m2-l2=0或H=constant的类时曲面定义了一个Darboux线汇,同时得到了相应的B(a)cklund变换.     

R~(2,1)中类时Weingarten曲面的Bcklund变换

本文利用平行变换，得到３维Ｍｉｎｋｏｗｓｋｉ空间Ｒ２，１中的Ｋ－２ｍＨ＋ｍ２－ｌ２＝０（Ｈ２＞ Ｋ，０≤ｍ＜ｌ）类时Ｗｅｉｎｇａｒｔｅｎ曲面的Ｂａｃｋｌｕｎｄ定理．     

关于类空W超曲面的一个存在性定理

利用超曲面的旋转对称性，将ＰＤＥ的求解转化为ＯＤＥ的求解，确定了Ｄｅ Ｓｉｔｔｅｒ空间中的一类旋转型的Ｗｅｉｎｇａｒｔｅｎ超曲面。即获得：给定Ｒ＾ｎ－１内开集（０，∞）＾ｎ－１上一个Ｃ＾１函数ｋｎ＝ｆ（ｋ１，…，ｋｎ－１）（ｎ≥２），一定存在Ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１内的ｎ维类空旋转超曲面Ｍ，使得Ｍ的ｎ个主曲率ｋ１，…，ｋｎ恰有上述函数关系。     

The Christoffel-Minkowski Problem Ⅱ: Weingarten Curvature Equations

In this paper the authors discuss the existence and convexity of hypersurfaces with prescribed Weingarten curvature.     

Introducing Weingarten cyclic surfaces in R3

In this paper, Cyclic surfaces are introduced using the foliation of circles of curvature of a space curve. The conditions on a space curve such that these cyclic surfaces are of type Weingarten surfaces or HK-quadric surfaces are obtained. Finally, some examples are given and plotted.     

Elliptic Weingarten hypersurfaces of Riemannian products

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of $M\times \mathbb{R}$, which are those whose principal curvatures $k_1,\cdots ,k_n$ and angle function $𝛩$ satisfy a relation $W(k_1,\cdots ,k_n,{𝛩}^2)=0$, being W a differentiable function which is symmetric with respect to $k_1,\cdots ,k_n$. When $\partial W/\partial k_i>0$ on the positive cone of ${\mathbb{R}}^n$, a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of $M\times \mathbb{R}$ which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either ${\mathbb{S}}^n\times \mathbb{R}$ or ${\mathbb{H}}^n\times \mathbb{R}$ is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$.     

R~(n+1)中的线性Weingarten超曲面的刚性定理

利用熟知的Cheng-Yau算子研究了欧氏空间R~(n+1)中的线性Weingarten超曲面,主要结果是有关标准球面的两个刚性定理.     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

The Paneitz Curvature Problem on Lower-Dimensional Spheres

In this paper we prescribe a fourth order conformal invariant (the Paneitz curvature) on the n-spheres, with n∊{5,6}. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results. Mathematics Subject Classifications (2000): 35J60, 53C21, 58J05, 35J30.     

Time-like Linear Weingarten Surfaces in Lorentzian Space Forms

In this paper we study Backlund transformations of time-like linear Weingarten surfaces with negative discriminant in Lorentzian space forms.     

Equivariance and Multiplicity for the Scalar Curvature Problem

Given (M, g ₀) a three-dimensional compact Riemannian manifold, assumed not to be conformally diffeomorphic to the standard unit 3-sphere, and G a compactsubgroup of the conformal group of (M, g ₀), we first study conditions for a smooth G-invariant function f to be the scalar curvature of a G-invariant conformalmetric to g ₀. Then, extending previous results of Hebeyand Vaugon, we study conditions for f to be the scalarcurvature of at least two conformal metrics to g ₀.     

A Stefan Problem with Curvature Correction in a Concentrated Capacity

We consider a Stefan problem with the curvature correction in a concentrated capacity. It is a kind of phase transition problems, in which the unknown temperature satisfies both the Stefan condition and the curvature correction on the free boundary, and also fulfills a kind of heat equations with special inner heat source. The inner beat source is related to the derivative of the temperature with respect to the direction vertical to the phase regions. The result established in this paper is the existence of a global weak solution of this problem.