具平行平均曲率向量的完备子流形(英文)

本文运用 Omori-Yau 的广义极大值原理,给出欧氏球面上全脐点子流形的一个充分条件,它是关于子流形的第二基本形武长度平方和平均曲率之间的一个不等式.     

局部对称Lorentz空间中的类空超曲面

研究了局部对称Lorentz空间中具有常平均曲率或常数量曲率的类空超曲面.利用丘成桐的广义极大值原理和自伴随算子得到了两个重要的内蕴刚性定理,其分别推广了欧阳崇祯和刘新民的主要结果.     

具混合边值条件的一类积分微分方程解的存在唯一性

利用极大单调算子和伪单调算子值域的一些结果,研究了一类含有广义p-Laplace算子的、具有混合边值条件的积分微分方程,得到了这个方程解的存在唯一性的结果.所用方法是对以往一些研究工作的推广和补充.     

含有广义p-Laplacian算子的双曲型非线性微分方程的单调性方法

利用伪单调算子和极大单调算子值域的扰动结果,得到了含有广义p-Laplacian算子、具混合边值条件的双曲型非线性微分方程存在唯一解的抽象结论,是对含有广义p-Laplaucian算子的非线性椭圆方程和抛物方程相关研究工作的推广.运用了一些新的证明技巧.而且,展示了这个唯一解与某极大单调算子零点之间的关系.     

连续广义框架及其对偶的刻画,冗余及扰动

利用算子方法研究连续广义框架及其对偶的一些性质.首先给出连续广义框架及其对偶的算子刻画;证明在合适的条件下在一个给定连续广义框架的基础上去掉部分元素后剩余的部分还能构成连续广义框架;最后得到连续广义框架及其对偶的一些扰动结果.     

Banach空间中闭线性算子广义预解式存在定理

在Banach空间中研究闭线性算子广义逆扰动问题和广义预解式存在性问题.给出了闭线性算子广义逆在T-有界扰动下的一些稳定特征,这些特征推广了在有界线性算子情形、闭线性算子有界扰动情形以及闭线性算子保值域或保核空间情形的一些已知结果.以此为基础,得到了闭线性算子广义预解式存在的一些充要条件及其广义预解式的显式表达式.作为应用,给出了闭Fredholm算子和闭半-Fredholm算子的广义预解式存在性特征.     

广义Schwarz不等式的一个应用

本文利用广义Schwerz不等式讨论非负算子幂的保序性,并得到一些有趣的结果.     

广义逆混合拟变分不等式解的存在性与误差界（英文）

本文引入并研究希尔伯特空间中一类新的广义逆混合拟变分不等式问题(GIMQVI).利用广义投影算子的性质,得到了GIMQVI解的存在性和唯一性结果,而且得到了利用剩余函数刻画的GIMQVI的误差界.本文得到的结果推广和改进了近期文献的一些结果.     

含有广义p-Laplace算子的非线性边值问题解的存在性的研究

该文研究了两类含有广义p-Laplace算子的非线性边值问题. 首先, 利用变分不等式解的存在性的结果, 证明了含有广义p-Laplace算子的非线性Dirichlet边值问题解的存在性. 然后, 提出了一类含有广义p-Laplace算子的非线性Neumann边值问题. 通过深入挖掘这两类非线性边值问题间的关系, 借助于极大单调算子值域的一个扰动结果, 证明了含有广义p-Laplace算子的非线性Neumann边值问题解的存在性. 文中采用了一些新的证明技巧,推广和补充了作者以往的一些研究工作.     

复射影空间中具有常数量曲率的完备全实子流形

本文研究了复射影空间中具有常数量曲率的完备全实子流形的问题.利用丘成桐的广义极大值原理和自伴随算子,获得了这类子流形的某些内蕴刚性定理.     

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.     

Generalized Maximum Principles and Stochastic Completeness for Pseudo-Hermitian Manifolds*

In this paper, the authors establish a generalized maximum principle for pseudoHermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudoHermitian manifolds are deduced. Moreover, they prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, they give some applications of these generalized maximum principles.     

An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we derive a sharp estimate for the supremum of the scalar curvature (or, equivalently, the infimum of the squared norm of the second fundamental form) of a constant mean curvature hypersurface with two principal curvatures immersed into a Riemannian space form of constant curvature. Our results will be an application of the generalized Omori-Yau maximum principle, following the approach by Pigola et al. (Memoirs Am Math Soc 822, 2005).     

A Bernstein type theorem in ? × ? n

In this paper, as suitable application of the so-called Omori-Yau generalized maximum principle, we obtain a Bernstein type theorem concerning to complete hypersurfaces immersed with constant mean curvature in the product space ℝ × ℍ ⁿ . Furthermore, we treat the case that such hypersurfaces are vertical graphs.     

Maximum Principles and Singular Elliptic Inequalities

In this paper, we present a version of the Omori-Yau maximum principle, a Liouville-type result, and a Phragmen-Lindelöff-type theorem for a class of singular elliptic operators on a Riemannian manifold, which include the p-Laplacian and the mean curvature operator. Some applications of the results obtained are discussed.     

Maximum principle for semi-elliptic trace operators and geometric applications

Based on ideas of L. Alías, D. Impera and M. Rigoli developed in [13], we present a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theorem of [13, Thm. 16 and 21], see Theorems 5 and 6.     

Maximum principles and gradient Ricci solitons

It is shown that the Omori-Yau maximum principle holds true on complete gradient shrinking Ricci solitons both for the Laplacian and the f-Laplacian. As an application, curvature estimates and rigidity results for shrinking Ricci solitons are obtained. Furthermore, applications of maximum principles are also given in the steady and expanding situations.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

On the Omori-Yau almost maximum principle

We introduce a new sufficient condition for the conclusion of the Omori-Yau almost maximum principle in terms of the existence of a special exhaustion function. This seems presenting perhaps the easiest proof. We also demonstrate that the existing sufficient conditions imply our sufficient condition.     

A Rigidity Result of Spacelike Self-Shrinkers in Pseudo-Euclidean Spaces

In this paper, the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.