Existence of free boundary minimal surfaces

Abstract

Free boundary minimal surfaces naturally appear in various contexts, including partitioning problems for convex bodies, capillarity problems for fluids, and extremal metrics for Steklov eigenvalues on manifolds with boundary. Constructing embedded free boundary minimal surfaces is challenging, especially in ambient manifolds like the Euclidean unit ball, which only allow unstable solutions. Min-max theory offers a promising avenue for existence results, albeit with the added complexity of requiring control over the topology of the resulting surfaces. This presentation will offer an overview of recent results and applications. (Based on joint works with Alessandro Carlotto, Giada Franz and David Wiygul).

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