This research will focus on applications of analysis to problems in geometry. Jerry Kazdan will investigate continuation properties of some elliptic operators. Christopher Croke will continue his work on geometric inequalities using techniques from integral geometry. Dennis DeTurck will carry out research on the curvature operator, the spectrum of the Laplacian and on existence theorems for some partial differential equations. Charles' Epstein's research in complex geometry will involve the study of global biholomorphic invariants of Cauchy Riemann manifolds. They will be joined by Wenxiong Chen who will continue his research into variational problems that arise in geometry. Kazdan's research is aimed at establishing uniqueness of continuation properties of elliptic operators which arise in geometric settings. This is to be modelled on known results about harmonic functions where the corresponding operator is the Laplacian. Croke will investigate inequalities relating volume and injectivity radius. He will also study various associated stability results. DeTurck's research involves the study of nonlinear partial differential equations which arise in geometric settings. These occur frequently when one prescribes various invariants of a physical system or tries to find those geometric structures for which such invariants are optimal. Epstein's work will include a study of the behavior of Kaehler-Einstein metrics in the neighborhoods of simple singularities. The aim here is to understand the effect of interior singularities on the values of certain invariants.
