Differential Geometry and Partial Differential Equations

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Abstract Proposal: DMS-9803341 Principal Investigator: Richard M. Schoen This proposal deals with three geometric variational problems. The first is the problem of minimizing the volume for lagrangian submanifolds of symplectic manifolds. This theory provides an approach to constructing special lagrangian submanifolds of Kahler-Einstein manifolds. The second problem is the study of harmonic maps which are equivariant with respect to general isometric actions of discrete groups on spaces of nonpositive curvature. This theory may be applied to study many general rigidity questions for both finite and infinite dimensional representations of discrete groups. The third problem concerns the variational problem for Einstein metrics, where the problem is to compute the Yamabe invariant in more generality, and to show that standard metrics achieve this min-max variational characterization. Minimization problems occur in many branches of mathematics and science. For example, linear programming concerns the problem of minimizing a function (such as cost) subject to a set of constraint inequalities, problems of navigation involve finding paths of least length on the earth's surface, and in continuum mechanics, the equilibrium position of an elastic membrane is determined among the infinitely many possible positions by the condition that the potential energy be as small as possible. This proposal deals with certain geometric variational problems, of which the first is the problem of minimizing a potential energy for mappings subject to the constraint that the mappings preserve the area. Such problems arise in nonlinear elasticity where the mapping represents the deformation of an elastic body. The problems also arise in geometry where one can use such minimizing configurations to understand complicated geometric spaces that arise in string theory (physics). The second part of this proposal deals with maps which minimize a potential energy subject to the conditi on that they are symmetric for a complicated symmetry group. For example, if you consider the curve of least length which surrounds a given area, you get a circle, while if you choose curves which surround regions whose translates (under a fixed symmetry group) fill up the plane, then the solution is typically a special type of hexagon (the regular hexagon of the honeycomb if the symmetry group is chosen suitably). The main goal is to use the symmetric minimizing maps to understand the possible symmetry groups and how they can arise in important geometric situations. The final part of the proposal deals with equilibrium solutions of the Einstein equations of General Relativity. The full Einstein equations may be thought of as describing the vibrations of the gravitational field which determines the geometry of spacetime. The corresponding equilibrium problem is important in geometry, and the third part of the proposal deals with the question of the extent to which equilibrium solutions can be expected to minimize the potential energy.