Disc stackings and their Morse index

Abstract

We construct free boundary minimal disc stackings, with any number of strata, in the three-dimensional Euclidean unit ball, and prove uniform, linear lower and upper bounds on the Morse index of all such surfaces. Among other things, our work implies for any positive integer k the existence of k-tuples of distinct, pairwise non-congruent, embedded free boundary minimal surfaces all having the same topological type. In addition, since we prove that the equivariant Morse index of any such free boundary minimal stacking, with respect to its maximal symmetry group, is bounded from below by (the integer part of) half the number of layers, it follows that any possible realization of such surfaces via an equivariant min-max method would need to employ sweepouts with an arbitrarily large number of parameters. This also shows that it is only for N = 2 and N = 3 layers that free boundary minimal disc stackings can be obtained by means of one-dimensional mountain pass schemes.

Keywords

• minimal surfaces
• free boundary problems
• morse index
• 53a10
• 49q05
• 58e12