Differential Geometry and Partial Differential Equations

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Professor Schoen is proposing research in the areas of Differential Geometry and General Relativity. He is proposing to study solutions of the constraint equations which are not time symmetric with an eye to the study of the general Penrose inequality as well as an analysis of the stability of the constraint manifold. The latter topic is important for understanding numerical stability for the vacuum Einstein equations. Professor Schoen is also proposing to study the construction of submanifolds which are calibrated by the special lagrangian calibrating form. He plans to apply ideas from geometric measure theory to this problem. He also plans to study hamiltonian stationary submanifolds in dimension greater than two; in particular, the hamiltonian stationary tangent cones will be studied. Professor Schoen intends to investigate stable minimal surfaces which remain stable under coverings with the hope of showing that these are holomorphic in general situations. Finally, he intends to study geodesic completeness properties of hypersurfaces in Minkowski space of constant Gauss-Kronecker curvature. Professor Schoen's project will lead to a better understanding of solutions of the Einstein equations of General Relativity. A better theoretical understanding is essential for the success of accurate numerical modeling of solutions. Numerical modeling is important for predicting the nature of the gravitational radiation which arises from dynamic situations, and NSF currently has a large project, LIGO, which is attempting to measure this radiation. He believes that the theoretical work of this project will be helpful for the numerics. The remainder of Professor Schoen's proposed work involves the use of geometric methods to understand the behavior of surface interfaces, such as soap films and soap bubbles, of varying dimension which arise in physical situations. These natural geometric objects can be used to describe subtle properties of the spaces in which they reside. These spaces arise in physical models such as string theory where they play a basic role.