Disc stackings and their Morse index

TL;DR

This paper constructs free boundary minimal disc stackings in the 3D unit ball with multiple layers, providing bounds on their Morse index and implications for their existence via min-max methods.

Contribution

It introduces new free boundary minimal disc stackings with arbitrary layers and establishes bounds on their Morse index, revealing limitations of certain min-max constructions.

Findings

Uniform bounds on Morse index for all stackings

Existence of multiple non-congruent minimal surfaces with same topology

Limitations of one-dimensional min-max schemes for N>3

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.