Geometric Variational Problems

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Abstract- DMS-0104007-Geometric Variational ProblemsThe principal investigator proposes to continue the study, joint with R. Schoen, of constrained variationalproblems for lagrangian cycles. In its most basic form the problem can be posed as follows: Consider a symplecticmanifold with a metric compatible with the symplectic form. Fix a homology class that can be represented by a lagrangian cycle. Find a lagrangian cycle that minimizes volume among all lagrangian cycles representing this class and derive optimal regularity of this cycle. In the case that the symplectic manifold is Kaehler,with Kaehler-Einstein metric, sufficient regularity of the minimizer implies that the minimizer is both lagrangian and minimal (zero mean curvature). If the first Chern class is negative such submanifolds could be unique, in a suitable sense,and then useful in understanding the geometry of the ambient manifold. If the Kaehler manifold is a Calabi-Yau manifold sufficient regularity implies that the minimizer is a calibrated submanifold, a special lagrangian submanifold.We propose to investigate the existence and regularity of this and related varitional problems and to study the consequences of these results on thegeometry of Kaehler-Einstein manifolds.A consequence of the proposal is an existence theorem for special lagrangian submanifolds of a Calabi-Yau manifold.This result is an essential part of the program proposed by Strominger-Yau-Zaslow for the geometric construction of "mirrorsymmetry". Mirror symmetry is one of the most interesting and important problems currently being studied in mathematics and theoretical physics. At its core it proposes a duality between a class of manifolds called Calabi-Yau manifolds. This duality allows computations to be performed on one manifold that yield a result for its "mirror". Thus computations that are otherwise extremely difficult can be achieved.The realization of mirror symmetry will effect such diverse subjects as algebraic geometry, differential geometry, topology, partial differential equations and string theory.The current interest in this subject helps bridge the gap between physics and mathematics. In two-dimensions our proposal has some close analogies to a well-known model problemin non-linear elasticity. The regularity theory developed here will shed light on the difficult regularity problems of that theory. Finally this problem is the first attempt to make a systematic study of a variational problem with a geometric constaint. This idea will have other important applications in geometry and its applications.