Partial differential equations for manifolds with boundary

Princeton University, Princeton NJ

Investigators

Abstract

This project will study geometrically-defined linear operators that enjoy certain invariance properties when the underlying geometric structure undergoes changes that distort the lengths of curves but not the angles between them. The spaces under investigation, called manifolds with boundaries, serve often as background spaces with special structures (e.g., conformal geometry, Cauchy-Riemann geometry). The aforementioned linear operators prescribe the change in the shape of space, change quantified in terms of certain curvature measurement by means of a differential equation that is typically nonlinear. A long-term goal of this study is to find conditions under which these curvature equations are solvable and to study the quality of the solution. Successful results in the proposed project will enrich the subject of conformal geometry by providing a large new family of examples of special Einstein spaces that are of interest to a wide audience of mathematicians, as well as to mathematical physicists. In addition to contributing directly to the field of conformal geometry, this project will help train a number of graduate students and postdocs, who will have the dual role of researchers and educators of the next generation of mathematicians. The extensive lecture course conducted by the principal investigators contributes to the mathematical education of scientists in several geographic areas. A basic tool for this project is a version of a well-known inequality, the Sobolev inequality, that controls the size of a function in terms of a measurement of its variation. Indeed, it is often the case that a sharp version of such an inequality is what is needed. The principal investigators intend to find such sharp inequalities for certain spaces by making use of traditional methods from the calculus of variations. A secondary objective of the project is to improve upon previous work of the principal investigators in order to obtain bounds for a family of conformally compact Einstein spaces, bounds that might eventually allow one to construct of a large family of such spaces, as predicted by physicists. A third objective is to broaden the conditions under which one can assert the validity of a strong maximum principle for fourth-order partial differential equations, a technical device that only became available because of prior work by the prinicipal investigators. Such a maximum principle will allow for a number of applications not only to geometric questions but also to general analytic questions about elasticity and the bending of plates.