Non-linear Analysis in Riemannian Geometry

Cornell University, Ithaca NY

Investigators

Abstract

Abstract Award: DMS-0306495 Principal Investigator: Jose F. Escobar Professor Escobar plans to continue his research in linear and non-linear partial differential equations and its applications to Riemmannian Geometry. He will continue his study of solutions to Yamabe equations on compact Riemannian manifolds. These are semilinear elliptic equations and in the case that the manifold has a boundary they satisfy a non-linear boundary condition. Applications of the existence theory for these equations are proposed to solve other problems in differential geometry and relativity. He also proposes to study Hamilton's Ricci flow on two and three dimensional manifols with boundary. In the two dimensional case this is a scalar non-linear evolution equation satisfying a non-linear boundary condition, while in the three dimensional case is a system of nonlinear equations with nonlinear boundary conditions. He will continue the study of the relation between the geometry of the manifold and the eigenvalues for the Steklov problem, which is the study of harmonic functions whose normal derivative is proportional to the function. The geometric objects that will be studied are the so-called Riemannian manifolds. These are spaces endowed with analytical structures, like the metric which provide us with a way to measure lengths and angles. It is natural to study deformations of these structures to realize what properties in the space remain stable under such perturbations. The description of all these deformations is usually governed by differential equations. The curvature tensor of a Riemmannian manifold ( a measure for the "non-euclideanness" of a Riemannian space) usually makes such equations non-linear, although as in physics, most of them are of variational nature. From the earliest days, conformal changes of metric ( multiplication of the metric by a positive function) have played an importntant role in surface theory. The equations proposed appear in the problem of conformal deformation of a Riemannian metric and in relativity. The Steklov problem appears in mathematical physics, conformal geometry, spinor geometry, minimal surfaces, partial differential equations and harmonic analysis. Geometry as well as topology involves the study of properties of the space. But whereas geometry focuses on properties of space that involves size, shape and measurement, topology concerns itself with the less tangible properties of relative position and connectedness. In recent years Hamilton's Ricci flow has become a fundamental tool in the study of geometry and toplogy of manifolds.