Mario Schulz
The case of the Euclidean unit ball as ambient manifold for free boundary minimal surfaces arises naturally in the study of extremal metrics for Steklov eigenvalues on surfaces with boundary and is
therefore of particular interest.
Yet even in this specific case many fundamental questions remain open.
Two of the most basic ones can be phrased as follows:
(1) Can a surface of any given topology be realised as an embedded free boundary minimal surface in the 3-dimensional Euclidean unit ball? We answer this question affirmatively for surfaces with connected boundary and arbitrary genus.
(2) When they exist, are such embeddings unique up to ambient isometry? We answer this question in the strongest negative terms by providing pairs of non-isometric free boundary minimal surfaces with the same
topology and symmetry group.
