Applications for Potential Theory to Geometric Analysis

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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.