Degenerate Microlocal Methods in Geometric Analysis

Stanford University, Stanford CA

Investigators

Abstract

Abstract of NSF Award DMS 0805529 (Rafe Mazzeo): The PI's research focuses on a number of problems in geometric analysis involving degenerate elliptic or parabolic equations on stratified spaces. One theme is the search for canonical metrics on the class of compact iterated cone-edge spaces; special low-dimensional cases include a new analysis of Riemann surfaces with conic singularities, and more significantly, some robust deformation results for three dimensional polyhedra in space forms. Closely related is a project to determine the deformation theory of noncompact Einstein spaces modeled on noncompact symmetric spaces of general rank; this is a generalization of the theory of Poincare-Einstein metrics which has played a prominent role in many recent papers in conformal geometry, and also in string theory, where it appears as part of the AdS/CFT correspondence. Another project concerns minimal submanifolds of these Poincare-Einstein spaces, and in particular, in three-dimensional convex cocompact hyperbolic manifolds. The goal here is to define and study the variational properties of the renormalized area functional on the space of properly embedded minimal surfaces with embedded asymptotic boundary curves; this seems to be a natural context for studying the Willmore functional from differential geometry for surfaces with boundary. Finally, the PI has also been studying geometric evolution equations on singular spaces. The first case in terms of geometric simplicity is the generalization of the curvature flow from the setting of closed embedded curves to that of networks of curves; the goals are to establish a good existence theory for the flow, to examine questions of nonuniqueness and to prove long-time existence to a Steiner network. To put this work into a broader context, most work in geometric analysis is in the setting of geometric objects which are smooth, i.e. do not have corners, edges or other singularities. However, spaces with singularities appear naturally and very frequently in geometry, physics and other applications, and it is natural to try to extend the methods and results of geometric analysis to this broader class of objects. However, the appropriate tools from analysis and partial differential equations do not exist in this generality, so a major part of the work needed is to extend these techniques to spaces with singularities. This has been a major endeavour of the PI throughout his career. The specific problems on which he is now working involve refined questions such as the existence and nature of optimal metrics, or shapes, on these singular spaces, and the study of evolution equations which continuously deform such a space into one of these optimal shapes. On smooth spaces, these questions have been some of the principal directions of research in geometric analysis, but their analogues in this singular setting have not been studied in any systematic way. Another problem involves the study of a renormalized version of the surface area of a class of optimal (minimal) surfaces inside a space with constant negative curvature. This is an action functional which has been studied intensively in string theory. One final part of this PI's work is of an educational nature: he is the founder and director of a residential summer program (SUMaC), held at his institution each year, which is directed toward highly motivated and talented high school students to encourage their continued study of mathematics. This program is now in its fourteenth year of operation.