Principal Investigator: Jon Wolfson
The project will investigate the question of the existence minimal lagrangian submanifolds in Kaehler-Einstein manifolds and special lagrangian submanifolds in Calabi-Yau manifolds using variational and deformation techniques and using mean curvature flow. The ``Lagrangian Hodge Conjecture'' states that given an integral lagrangian homology class in a Calabi-Yau manifold there is a special lagrangian cycle that represents this class. This conjecture is now known to be false in a Calabi-Yau surface (a K3 surface). The failure of the conjecture is closely related to the existence of singularities (other than branch points) on minimizing lagrangians. This project aims to answer a related conjecture, the ``Deformation Lagrangian Hodge Conjecture'': Given a one-parameter family of Calabi-Yau metrics whose associated Kaehler forms have fixed cohomology class and given a lagrangian homology class, suppose that the initial metric in the family admits a special lagrangian cycle that represents the lagrangian homology class. Then, we conjecture, that the metrics in the one-parameter family sufficient close to the initial metric also admit special lagrangian cycles that represents this class. This conjecture has been established in the two dimension case (the case of special lagrangian surfaces in K3 surfaces) and in the case that the initial special lagrangian is an immersion. It is known that the mean curvature flow of a smooth lagrangian submanifold preserves the lagrangian condition if the ambient manifold is Kaehler-Einstein. The project intends to investigate various questions about the behavior of lagrangians under this flow, in particular, to study examples that can be used either to give counter-examples to the Thomas-Yau conjecture or show that any such counter-example must be global.
Variational problems with geometric constraints, deformation problems and mean curvature flow in codimension greater than one are on the frontier of mathematical analysis. These problems are natural in geometry but they are also important in many different applied problems. For example, in material science a model problem asks to find a minimizer of ``kinetic energy'' among area preserving maps between disks and to find the optimal smoothness of the minimizer. Parts of this project are closely related to this kind of existence and regularity question and therefore to developing techniques at the foundations of the theory of non-linear elasticity. In string theory, a branch of theoretical physics, well known work conjectures the existence of a certain class of volume minimizing three dimensional surfaces called special lagrangian submanifolds. This project is a direct attempt to resolve this conjecture. Mean curvature flow in various codimensions models many different physical phenomena, including, for example, flame propagation. This project attempts to exploit the ``lagrangian'' constraint to get an understanding of mean curvature flow in higher codimensions. The techniques investigated in this project hold the promise of enhancing the interaction between geometry and various fields of applied mathematics and engineering and in bringing new results and techniques into these fields.
