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Eigenvalue estimates and a compactness theorem for embedded minimal surfaces in the unit ball

University of British Columbia

Recently, A. Fraser and R. Schoen introduced an extremal eigenvalue problem for surfaces with boundary. They discovered an interesting relationship between extremal metrics and minimal surfaces in the unit ball satisfying a free boundary condition. In this talk, we will begin with some known results on the Steklov eigenvalue problem for the Dirichlet-to-Neumann map on compact manifolds with boundary. After giving some examples of minimal surfaces in the unit ball, we will prove a lower bound on the first Steklov eigenvalue for those which are embedded. We will then use this to show that the space of embedded minimal surfaces in the unit ball, with fixed topological type, is compact in the smooth topology. All these results apply if the ambient space is any compact 3-manifolds with nonnegative Ricci curvature and strictly convex boundary. In the end, we will mention some open problems in this direction. This is joint work with A. Fraser.

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