The project will investigate the question of the existence of special lagrangian submanifolds in Calabi-Yau manifolds using variational techniques and using mean curvature flow. The variational techniques involve the study of constrained variational problems for lagrangian cycles. These problems can be formulated in arbitrary Kaehler (or symplectic) manifolds and involve minimizing volume among lagrangians that represent a fixed homology class and proving optimal regularity of the minimizer. In particular the project hopes to show that if the ambient manifold is a Calabi-Yau $3$-fold then, for a suitably formulated problem, the minimizer is special lagrangian. It has long been known that the mean curvature flow of a lagrangian submanifold preserves the lagrangian condition if the ambient manifold is Kaehler-Einstein. The project intends to investigate the regularity properties of the mean curvature flow of a lagrangian submanifold of a Kaehler-Einstein manifold. In particular, it intends to investigate the conditions under which the flow does and does not develop singularities in finite time.
Variational problems with geometric constraints 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. 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. At present, the existence of a minimizer is known but nothing is known about its singularities. Parts of this project are closely related to this kind of ``regularity'' question. In string theory 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 answer this question in the affirmative. Mean curvature flow in various codimensions models many different physical phenomena. This project attempts to exploit the ``lagrangian'' constraint to get an understanding of mean curvature flow in higher codimensions. This subject is relatively unexplored. 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.
