Applications for Potential Theory to Geometric Analysis

建议DMS-0406504题目：位势理论在几何分析中的应用。PI：加州大学圣巴巴拉分校的Denis A.Labutin。该项目致力于研究黎曼几何中出现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。投影的中心问题是几何问题中出现的奇异集的分析。建议使用非线性位势理论的思想来处理几个问题。让我们来描述一下这些问题。奇异的Yamabe问题起源于Loewner和Nirenberg以及Schoen和Yau的工作。它包括寻找黎曼流形(单位球面为模型情形)上的度量到具有常数量曲率的完备度量的共形变形。问题是如何描述它可能的区域。在单位球面上的常负数量曲率的共形变形的情况下，最近被PI解决了。答案是，当且仅当余项在位势理论意义下不薄时，这是可能的。这意味着具有一定容量的维纳型测试在补码的任意点都成立。在零标量曲率的共形变形的情况下，有强有力的证据表明，判据将是补数相对于另一个容量的极性。PI打算验证这一点。我们能把结果从球面推广到一般的闭流形吗？在什么额外的假设下？位势理论的思想能导致正标量曲率的变形吗？另一组问题与负曲率Cartan-Hadamard流形上的Liouville定理有关。主要问题很容易说明。有一致上负曲率边界的大于3维的流形支持非平凡有界调和函数吗？Pi相信位势理论的思想有助于更好地理解这个问题。强负弯曲Cartan-Hadamard流形在无穷远处的Dirichlet问题的可解性(从而证明了Liouville定理的失败)是由Sullivan，Anderson，Schoen和Anconn建立的。它与平面空间中不规则区域上的Dirichlet问题有一定的相似之处，势能理论的思想被证明是有用的。Liouville定理在一个区域中的有效性被认为等同于补关于经典静电容量的极性。我们能为流形建立类似的关系吗？曲率是度量的充分特征吗？Grigoryan和Saloff-Costay的结果表明，对于Harnack不等式有效性这一不同但相关的问题，正确的语言是黎曼体积增长和局部Poincare型不等式，而不是非负曲率。可以肯定的是，新的方法将不得不发展到Liouville定理。该项目的主要目标是使用以前没有被应用于此类问题的技术来调查所描述的问题。这些都是非线性位势理论的技术。非线性偏微分方程组与几何之间的相互作用领域正在经历着强劲的发展。然而，目前的研究并不是主要针对项目中提出的问题。重点介绍了从非线性偏微分方程的技术难点领域，即非线性位势理论出发的方法在黎曼几何中出现的几个具体问题上的应用。以前，这些方法并没有在这样的背景下系统地应用和发展。更全面地理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点。对几何对象奇点的更好的理解反过来又会导致理论物理、拓扑学和其他数学领域的问题的进步。

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.