Abstract:

In the early 90's Yong-Geun Oh introduced the problem of studying critical points of the area among Lagrangian surfaces in a symplectic riemannian manifold. Such surfaces are called Hamiltonian stationary or sometimes H-minimal surfaces. This variational problem  is motivated by natural questions such as the study of the Plateau problem in Lagrangian homology classes, the construction of calibrated minimal surfaces in Calabi Yau geometry, the minmax construction of minimal surfaces in spheres....etc 

We will first present the difficulties with dealing with the Hamiltonian Stationary equation in general and present the construction of ``pathological solutions'' to this equation in 2 dimension which are nowhere continuous. Then we will turn to the special case of area minimizing H-minimal surfaces and the discovery in the early 2000 of a family of singularities of conical type by Schoen and Wolfson.

We will present a direct method for constructing Hamiltonian Stationary discs with prescribed Schoen Wolfson cones.

If time permits we will introduce a weak formulation of the H-minimal equations excluding a-priori pathological solutions and compatible with min-max variational constructions.

We will then formulate a conjecture relative to the singular set of general solutions to this new variational formulation of H-minimal surfaces.

For further information please contact elisur.magrini@unibocconi.it