Linear and Non-Linear Eigenvalues in Geometry

Cornell University, Ithaca NY

Investigators

Abstract

Abstract Award: DMS-0072164 Principal Investigator: Jose F. Escobar Professor Escobar proposes to work in three different variational problems: The first one is on conformal deformation of metrics. He proposes to work on the scalar curvature problem on scalar flat manifolds of dimension five or more. In addition, he will study the prescribed scalar curvature and prescribed mean curvature problem on manifolds with boundary; particular attention will be given to this problem when the manifold is the Euclidean ball and the dimension is three. Earlier investigations indicate that the problem in three dimensions is special. The second topic he proposes to study is estimates for the first non-zero Steklov eigenvalue on compact manifolds with boundary. Escobar proposes to study relations between the geometry of the space and the first non-zero eigenvalue and apply this information to problems in conformal geometry, heat flow problems, and to the study of eigenvalues of minimal surfaces. The third topic is to study Einstein metrics on manifolds with boundary. There are three different kinds of equations that arise naturally as a variational problem of a functional introduced by the proposer; they are Einstein metrics satisfying that the boundary is totally geodesic or, more generally, that the boundary is umbilic, and Ricci flat metrics with umbilic boundary. The three problems above have their roots in Riemannian geometry as well as in physics. The Steklov problem initially appeared in physics, then in harmonic analysis, partial differential equations, conformal geometry, and minimal surfaces. In physics, it describes the temperature of a body where the flux through out the boundary is proportional to the temperature. We will investigate how the geometry of the space influence the first non-zero eigenvalue, that is, the smallest nonzero constant of proportionality. The Einstein equation proposed in this project is the generalization of the Einstein's equation in boundaryless spaces studied by Hilbert and Einstein in general relativity to the case of spaces with boundary. The boundary conditions we will imposed are the natural ones if one studies this problem from the point of view of the calculus of variations. The scalar curvature equations that we will investigate are the average version of the Einstein equation on manifolds with boundary. Nowadays they are known as the Yamabe type equations. These equations appear in relativity and in other branches of physics.