Spectral estimates for free boundary minimal surfaces via Montiel-Ros partitioning methods

TL;DR

This paper extends spectral estimation methods to free boundary minimal surfaces with boundary, incorporating group actions and boundary conditions, and provides bounds on the Morse index related to surface topology.

Contribution

It adapts Montiel-Ros techniques to manifolds with boundary and symmetry, deriving bounds on Morse index and nullity for free boundary minimal surfaces.

Findings

Computed exact equivariant Morse index and nullity for two surface families.

Derived explicit linear bounds on Morse index when symmetry constraints are removed.

Extended spectral estimate methods to include mixed boundary conditions and group actions.

Abstract

We adapt and extend the Montiel-Ros methodology to compact manifolds with boundary, allowing for mixed (including oblique) boundary conditions and also accounting for the action of a finite group $G$ together with an additional twisting homomorphism $\sigma\colon G\to\operatorname O(1)$. We then apply this machinery in order to obtain quantitative lower and upper bounds on the growth rate of the Morse index of free boundary minimal surfaces with respect to the topological data (i. e. the genus and the number of boundary components) of the surfaces in question. In particular, we compute the exact values of the equivariant Morse index and nullity for two infinite families of examples, with respect to their maximal symmetry groups, and thereby derive explicit two-sided linear bounds when the equivariance constraint is lifted.