Curvature flows and spectral estimates.

Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, the applicant and collaborators have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this program: it will produce new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which noncompact solutions to curvature flows are stable.