Topics in Differential Geometry

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AbstractAward: DMS 0604759Principal Investigator: Jon WolfsonThe project will investigate the question of the existence minimal lagrangiansubmanifolds in Kaehler-Einstein manifolds and special lagrangian submanifoldsin Calabi-Yau manifolds using variational and deformation techniques and usingmean curvature flow. The ``Lagrangian Hodge Conjecture'' states that given anintegral lagrangian homology class in a Calabi-Yau manifold there is a speciallagrangian cycle that represents this class. This conjecture is now known tobe false in a Calabi-Yau surface (a K3 surface). The failure of the conjectureis closely related to the existence of singularities (other than branchpoints) on minimizing lagrangians. This project aims to answer a relatedconjecture, the ``Deformation Lagrangian Hodge Conjecture'': Given aone-parameter family of Calabi-Yau metrics whose associated Kaehler forms havefixed cohomology class and given a lagrangian homology class, suppose that theinitial metric in the family admits a special lagrangian cycle that representsthe lagrangian homology class. Then, we conjecture, that the metrics in theone-parameter family sufficient close to the initial metric also admit speciallagrangian cycles that represents this class. This conjecture has beenestablished in the two dimension case (the case of special lagrangian surfacesin K3 surfaces) and in the case that the initial special lagrangian is animmersion. It is known that the mean curvature flow of a smooth lagrangiansubmanifold preserves the lagrangian condition if the ambient manifold isKaehler-Einstein. The project intends to investigate various questions aboutthe behavior of lagrangians under this flow, in particular, to study examplesthat can be used either to give counter-examples to the Thomas-Yau conjectureor show that any such counter-example must be global.Variational problems with geometric constraints, deformation problems and meancurvature flow in codimension greater than one are on the frontier ofmathematical analysis. These problems are natural in geometry but they arealso important in many different applied problems. For example, in materialscience a model problem asks to find a minimizer of ``kinetic energy'' amongarea preserving maps between disks and to find the optimal smoothness of theminimizer. Parts of this project are closely related to this kind ofexistence and regularity question and therefore to developing techniques atthe foundations of the theory of non-linear elasticity. In string theory, abranch of theoretical physics, well known work conjectures the existence of acertain class of volume minimizing three dimensional surfaces called speciallagrangian submanifolds. This project is a direct attempt to resolve thisconjecture. Mean curvature flow in various codimensions models many differentphysical phenomena, including, for example, flame propagation. This projectattempts to exploit the ``lagrangian'' constraint to get an understanding ofmean curvature flow in higher codimensions. The techniques investigated inthis project hold the promise of enhancing the interaction between geometryand various fields of applied mathematics and engineering and in bringing newresults and techniques into these fields.