Free boundary minimal surfaces with three boundary components

The symmetry and topology of a free boundary minimal surface in the Euclidean unit ball do not determine the surface uniquely. There exist infinitely many pairs of non-isometric free boundary minimal surfaces having the same genus g, three boundary components and antiprismatic symmetry group of order 4(g+1).

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Free boundary minimal surfaces with four boundary components via doubling the critical catenoid

A doubling of the the critical catenoid can be glued to an embedded free boundary minimal surface with sufficiently large, odd genus and four boundary components.

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Free boundary analogues of the higher order Costa–Wohlgemuth surfaces

Conjecturally, the Euclidean unit ball also contains free boundary minimal surfaces which look like the higher order Costa–Wohlgemuth surfaces, i.e. they have sufficiently large, even genus and four boundary components.