The project is dedicated to investigation of main elliptic partial differential equations arising in Riemannian geometry, namely equations with the Monge-Ampere operator, Laplacian, and conformal Laplacian. The central question for the project is the analysis of the singular sets arising in geometric problems. It is proposed to approach several problems using ideas from nonlinear potential theory. Let us describe the problems. The singular Yamabe problem originates in the work of Loewner and Nirenberg, and Schoen and Yau. It consists of finding conformal deformation of the metric 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 negative scalar curvature in the unit sphere it was recently solved by PI. The answer is that it is possible if and only if the complement is not thin in the potential theory sense. This means that the Wiener-type test with a certain capacity holds at any point of the complement. In the case of the conformal deformation to the zero scalar curvature there is a strong evidence that the criterion will be the polarity of of the complement with respect to another capacity. PI intends to verify it. Can one extend the results from the sphere to general closed manifolds? Under what additional assumptions? Can potential theory ideas contribute to the deformation to positive scalar curvature? Another group of questions is related to the Liouville theorems on negatively curved Cartan-Hadamard manifolds. The main problem is easy to state. Does any such manifold of dimension greater than three with uniform upper negative sectional curvature bound support a nontrivial bounded harmonic function? PI believes that potential theory ideas can contribute to better understanding of this question. The solvability of the Dirichlet problem at infinity (and hence the failure of the Liouville theorem) for strongly negatively curved Cartan-Hadamard manifolds was established in the works by Sullivan, Anderson, Schoen, and Ancona. There are certain similarities with the Dirichlet problem in irregular domains in the flat space, where potential theory ideas are proved to be useful. Validity of Liouville theorems in a domain is known to be equivalent to the polarity of the complement with respect to the classical electrostatic capacity. Can one establish a similar relation for the manifolds? Is curvature the adequate characteristic of the metric for such problems? Results of Grigoryan and Saloff-Coste show, that for a different but related question of the validity of Harnack inequality, the correct language is the Riemannian volume growth and local Poincare-type inequalities rather than the nonnegative curvature. It is certain that the new methods will have to be developed for Liouville theorems. The main goal of the project is investigation of the described problems using the techniques which have not been applied to such problems before. These are the techniques of nonlinear potential theory.
The area of interaction between nonlinear partial differential equations and geometry is undergoing a strong development. However, the current research is not primarily aimed at the questions proposed in the project. The project is focused on the application of the methods from technically difficult area of nonlinear partial differential equations, namely nonlinear potential theory, to several concrete problems about singularities arising in Riemannian geometry. Previously these methods were not systematically applied and developed in such context. More complete understanding of potential theoretic methods in geometric analysis will be of great help in detailing the way to better understanding of singularities in geometric problems. Better understanding of singularites of geometric objects leads in its turn to progress in problems from theoretical physics, topology and other areas of mathematics.
