|AILANA FRASER, Courant Institute, New York, New York 10012, USA|
|On the free boundary variational problem for minimal disks|
We will discuss the problem of extremizing the energy (equivalently area) for maps from the unit disk D into a Riemannian manifold Nhaving boundary lying on a specified embedded submanifold M. The critical points of this geometric variational problem are minimal surfaces which meet the submanifold orthogonally along the boundary. We derive a partial Morse theory for this problem in arbitrary dimensions. In addition, we use the geometry to obtain certain lower bounds on the Morse index for such disks.