| In this thesis we consider the problem of extremizing the energy (equivalently area) for maps from the unit disk D into a Riemannian manifold {dollar}Nsp{lcub}n{rcub}{dollar} having boundary lying on a specified embedded submanifold {dollar}Msp{lcub}m{rcub}{dollar}. The critical points of this geometric variational problem are minimal surfaces which meet the submanifold orthogonally along the boundary. This free boundary problem is well studied classically in {dollar}IRsp3{dollar} in case one seeks an area minimizing disk. In this thesis we consider the more general problem of finding solutions which are predicted by Morse theory based on the topology of the mapping space. We construct solutions with Morse index at most k under the assumption that the corresponding space of continuous maps has nontrivial homotopy in dimension k. This can in turn be related to the relative homotopy of the pair {dollar}(N, M){dollar}.; In addition to extending the minimal surface theory, our goal is also to apply this theory in Riemannian geometry in a spirit analogous to the second variation theory for geodesics. In order to accomplish this it is important to understand the relations between the curvature of N, the second fundamental form of M, and the Morse index of solutions. We prove instability of solutions in the case where N is a convex domain in {dollar}IRsp{lcub}n{rcub}{dollar} and the constraint submanifold {dollar}M = partial N{dollar}. More generally, for "two-convex" domains, we also get a lower bound on the Morse index of solutions. These estimates hold in the case of curved ambient manifolds under the assumption of positive isotropic curvature, when N is two-convex. In the special case where the domain is the standard ball in {dollar}IRsp{lcub}n{rcub}{dollar}, we prove instability of solutions of the free boundary problem in any dimension and codimension. |
