Applications for Potential Theory to Geometric Analysis

势理论在几何分析中的应用

• 批准号: 0406504
• 负责人: Denis Labutin
• 金额: $ 9.2万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406504&HistoricalAwards=false
• 关键词: Applications; Potential; Theory; Geometric; Analysis

Proposal DMS-0406504Title: Applications of potential theory to geometric analysis.PI: Denis A. Labutin, University of California, Santa BarbaraABSTRACTThe project is dedicated to investigation of main ellipticpartial differential equations arising inRiemannian geometry, namely equations with the Monge-Ampere operator,Laplacian, and conformal Laplacian. The central question for the projectis the analysis of the singular sets arising in geometric problems.It is proposed to approach several problems using ideasfrom nonlinear potential theory. Let us describe the problems.The singular Yamabe problem originates in the work ofLoewner and Nirenberg, and Schoen andYau. It consists of finding conformal deformation of themetric in a domain of a Riemannian manifold (unit sphere is the model case)to a complete metric with a constant scalar curvature.The question is how to describe the domains for which it is possible.In the case of the conformal deformation to the constant negativescalar curvature in the unit sphere it was recently solved by PI.The answer is that it is possible if and only if the complementis not thin in the potential theory sense. This means that theWiener-type test with a certain capacity holds at any point ofthe complement. In the case of the conformal deformation to the zero scalar curvaturethere is a strong evidence that the criterion will be the polarity ofof the complement with respect to another capacity.PI intends to verify it. Can one extend the resultsfrom the sphere to general closed manifolds?Under what additional assumptions?Can potential theory ideas contributeto the deformation to positive scalar curvature?Another group of questions is related to theLiouville theorems on negatively curvedCartan-Hadamard manifolds. The main problem is easy to state. Does any such manifoldof dimension greater than three with uniform upper negative sectional curvature boundsupport a nontrivial bounded harmonic function?PI believes that potential theory ideascan contribute to better understandingof this question. The solvability of the Dirichlet problemat infinity (and hence the failure of the Liouville theorem)for strongly negatively curved Cartan-Hadamard manifoldswas established in the works bySullivan, Anderson, Schoen, and Ancona.There are certain similarities with the Dirichlet problemin irregular domains inthe flat space, wherepotential theory ideas are proved to be useful.Validity of Liouville theorems in a domainis known to be equivalent to the polarity ofthe complement with respect to the classicalelectrostatic capacity. Can one establish a similar relation forthe manifolds? Is curvature the adequatecharacteristic of the metric for such problems?Results of Grigoryan and Saloff-Costeshow, that for a different but related question of the validity ofHarnack inequality, the correct language is the Riemannian volume growthand local Poincare-type inequalities rather than the nonnegative curvature.It is certain that the new methods will have to be developed forLiouville theorems. The main goal of the project is investigation ofthe described problems using the techniques which have not been applied to such problemsbefore. These are the techniques of nonlinear potential theory.The area of interaction between nonlinear partial differential equationsand geometry is undergoing a strong development. However, the current researchis not primarily aimed at the questions proposed in the project. The projectis focused on the application of the methods fromtechnically difficult area of nonlinear partial differential equations,namely nonlinear potential theory, to several concrete problems aboutsingularities arising in Riemannian geometry. Previously these methods were notsystematically applied and developed in such context. More complete understanding ofpotential theoretic methods in geometric analysis will be of great help indetailing the way to better understanding of singularities in geometric problems.Better understanding of singularites of geometric objects leads in its turn toprogress in problems from theoretical physics, topology and other areas of mathematics.

提案DMS-0406504标题：势理论在几何分析中的应用。PI：Denis A.实验室，加州大学圣巴巴拉分校摘要该项目致力于研究黎曼几何中的主要椭圆型偏微分方程，即具有Monge-Ampere算子、拉普拉斯算子和共形拉普拉斯算子的方程。该项目的中心问题是分析几何问题中出现的奇异集，提出用非线性势理论的思想来处理几个问题。奇异Yamabe问题起源于Loewner和Nirenberg以及Schoen和Yau的工作。它是在黎曼流形的一个区域（单位球面是模型情形）上求出度量到具有常数量曲率的完备度量的共形变形.问题是如何描述可能的区域.在单位球面上求出度量到具有常负数量曲率的完备度量的共形变形的情况下， 最近PI解决了这个问题。答案是，当且仅当在势理论的意义上互补不薄时，这是可能的。这意味着具有一定容量的维纳型检验在补码的任何点都成立。在保形变形为零标量曲率的情况下，有一个强有力的证据表明，该准则将是关于另一个容量的补的极性。PI打算验证它。可以将结果从球面推广到一般闭流形吗？在什么样的假设下？势理论的思想能促成正标量曲率的变形吗？另一类问题与负曲率Cartan-Hadamard流形上的Liouville定理有关。 主要的问题很容易陈述。任何具有均匀上负截面曲率边界的大于三维的流形是否支持非平凡有界调和函数？PI认为，潜在的理论思想可以有助于更好地理解这个问题。 无穷远处Dirichlet问题的可解性Sullivan，安德森，Schoen和安科纳等人的工作中建立了强负曲Cartan-Hadamard流形上的Dirichlet问题（并因此导致Liouville定理的失败）。其中势理论的思想被证明是有用的。刘维尔定理在一个域中的有效性被认为是等价于补的极性关于经典静电容量我们能为流形建立一个类似的关系吗？ 曲率是度量的适当特征吗？Grigoryan和Saloff-Coste的结果表明，对于Harnack不等式的有效性这一不同但相关的问题，正确的表述是Riemann体积增长和局部Poincare型不等式，而不是非负曲率.该项目的主要目标是调查所描述的问题，使用的技术还没有被应用到这样的问题之前。 非线性偏微分方程与几何学之间的相互作用领域正在经历一个强有力的发展。然而，目前的研究主要不是针对项目中提出的问题.本项目着重于将非线性偏微分方程的技术难点--非线性位势理论的方法应用于黎曼几何中有关奇点的几个具体问题。以前，这些方法并没有在这种背景下系统地应用和发展。更全面地了解几何分析中潜在的理论方法将有助于更好地理解几何问题中的奇异性，更好地理解几何对象的奇异性进而导致理论物理、拓扑学和其他数学领域问题的进展。