One aspect of the proposed research has to do with optimal shapes of surfaces. If we think of the surface as a drumhead which vibrates freely at certain frequencies, then, roughly speaking, the more complicated a geometry we have the smaller its fundamental frequencies will be. This suggests the problem of looking for geometries which maximize the fundamental frequency for their area. This extremal question is a difficult and much studied problem. It turns out that the geometries which arise are related to surfaces of least area (soap films). The proposer will investigate such extremal configurations for surfaces with boundary. The other main area of investigation concerns the Einstein equations of general relativity. These equations describe the gravitational field for massive bodies in the universe. The theory is purely geometric and is a wave theory with an initial value formulation. The proposer is planning to investigate the geometry of solutions to give conditions under which gravitational collapse takes place and black holes are formed. Such questions lead to important geometric questions involving gravitational energy and curvature of spacetime.
The proposed research is at the interface between differential geometry, general relativity, and partial differential equations. A main theme of the research in geometry will be the study of spectral geometry. The PI plans to construct metrics on surfaces and certain higher dimensional manifolds subject to an area or boundary length constraint which maximize the first eigenvalue. This is a nonstandard type of variational problem since it involves maximizing and minimizing over infinite dimensional spaces of competitors. The PI will study the geometry of such maximizing metrics to determine the optimal shapes with largest fundamental frequency. In relativity, the PI intends to continue his investigations into the construction proposed by Bartnik of mass minimizing extensions of compact domains and static vacuum metrics. The PI also intends to study geometric properties of initial data sets concerning the question of whether they can contain non-compact stable trapped surfaces. Finally the PI intends to investigate global properties of the moduli space of solutions of the constraint equations which define the possible initial data for the Einstein equations. The PI shall pursue a range of questions concerning minimal submanifolds satisfying free boundary conditions and connections to eigenvalue problems. Finally the PI plans to continue his study of minimal lagrangian and special lagrangian submanifolds of Kahler-Einstein manifolds. The research will attempt to prove a conjecture concerning the invariance of the subgroup of the integral homology of a Calabi-Yau manifold which is generated by minimal lagrangian cycles when one deforms the ambient Calabi-Yau structure.
