Proposal: DMS 9802487 Principal Investigator: Jon Wolfson
In this proposal we will continue our study, joint with R. Schoen, of variational problems for lagrangian submanifolds: Let M be a symplectic 2n-manifold equipped with a compatible metric. Given a homology class in M that can be represented by an immersed lagrangian submanifold, find a lagrangian that minimizes volume among all lagrangians that represent this homology class. The minimizer is known to exist as a lagrangian rectifiable varifold, so the focus of our research is on the regularity of these minimizers. When the ambient manifold M is Kaehler-Einstein it may be possible to show that the lagrangian minimizers are stationary, in the classical sense. We hope to develop a good regularity theory of lagrangian minimizers and then to use these minimizers to study the geometry of the ambient Kaehler or symplectic manifolds.
Many problems in physics and mathematics have solutions that are obtained via an optimization procedure. In this project we will study a minimizing procedure in differential geometry. We minimize volume among generalized surfaces (immersed submanifolds) that satisfy a geometric constraint (that are lagrangian). Because we require our generalized surfaces to satisfy this constraint many new difficulties arise. However our solutions often have interesting and important geometric properties. This problem has many features in common with problems in nonlinear elasticity. It is hoped that our techniques will be of use in that subject. However it should be emphasized that optimization procedures in the presence of geometric constraints have not been studied before. This idea will eventually be useful in many different contexts.
