Linear and Non-Linear Eigenvalues in Geometry

点击翻译按钮获取中文摘要

AbstractAward: DMS-0072164Principal Investigator: Jose F. EscobarProfessor Escobar proposes to work in three different variationalproblems: The first one is on conformal deformation ofmetrics. He proposes to work on the scalar curvature problem onscalar flat manifolds of dimension five or more. In addition, hewill study the prescribed scalar curvature and prescribed meancurvature problem on manifolds with boundary; particularattention will be given to this problem when the manifold is theEuclidean ball and the dimension is three. Earlierinvestigations indicate that the problem in three dimensions isspecial. The second topic he proposes to study is estimates forthe first non-zero Steklov eigenvalue on compact manifolds withboundary. Escobar proposes to study relations between thegeometry of the space and the first non-zero eigenvalue and applythis information to problems in conformal geometry, heat flowproblems, and to the study of eigenvalues of minimalsurfaces. The third topic is to study Einstein metrics onmanifolds with boundary. There are three different kinds ofequations that arise naturally as a variational problem of afunctional introduced by the proposer; they are Einstein metricssatisfying that the boundary is totally geodesic or, moregenerally, that the boundary is umbilic, and Ricci flat metricswith umbilic boundary.The three problems above have their roots in Riemannian geometryas well as in physics. The Steklov problem initially appeared inphysics, then in harmonic analysis, partial differentialequations, conformal geometry, and minimal surfaces. In physics,it describes the temperature of a body where the flux through outthe boundary is proportional to the temperature. We willinvestigate how the geometry of the space influence the firstnon-zero eigenvalue, that is, the smallest nonzero constant ofproportionality. The Einstein equation proposed in this projectis the generalization of the Einstein's equation in boundarylessspaces studied by Hilbert and Einstein in general relativity tothe case of spaces with boundary. The boundary conditions wewill imposed are the natural ones if one studies this problemfrom the point of view of the calculus of variations. The scalarcurvature equations that we will investigate are the averageversion of the Einstein equation on manifolds with boundary.Nowadays they are known as the Yamabe type equations. Theseequations appear in relativity and in other branches of physics.