A strong topological non-uniqueness result for free boundary minimal surfaces

Author: Alessandro Carlotto

Abstract:The study of free boundary minimal surfaces (namely: of critical points for the area functional in the category of relative cycles), which goes back at least to Courant, has played a distinguished role within the class of geometric variational problems for almost a century.
Yet, several fundamental questions remain open. For instance, is it possible to realise any orientable, compact surface with boundary as a free boundary minimal surface in the Euclidean unit ball? And, if so, are such realisations unique modulo ambient isometries? I will present significant advances on these two questions, including a (very recent) strong non-uniqueness result. In joint work with M. Schulz and D. Wiygul, we showed that the topology and symmetry group of a free boundary minimal surface in the Euclidean unit ball do not determine the surface uniquely: for any sufficiently large integer g there exist in the unit ball two distinct, properly embedded, free boundary minimal surfaces having genus g, three boundary components and symmetry group coinciding with the antiprismatic group of order 4(g+1).