On the Index of Fraser-Sargent-type minimal surfaces

TL;DR

This paper analyzes the Morse index and nullity of Fraser-Sargent minimal surfaces, providing bounds and numerical evidence of stability, and introduces conjectures for improved index estimates.

Contribution

It computes the Morse index and nullity of extended Fraser-Sargent surfaces and offers new bounds and conjectures on their stability properties.

Findings

Computed Morse index and nullity for extended Fraser-Sargent surfaces.

Provided numerical evidence suggesting stability of exterior parts.

Established lower and upper bounds on the index, with conjectures for improvements.

Abstract

Fraser-Sargent surfaces are free boundary minimal surfaces in the four-dimensional unit Euclidean ball. Extended infinitely they define immersed minimal surfaces in the Euclidean space. In the present paper we compute the Morse index and the nullity of these extended minimal surfaces. The parts of these surfaces outside the ball are exterior free boundary minimal surfaces. We provide a numerical evidence that they are stable. As a corollary of these results we obtain a lower bound on the index of Fraser-Sargent surfaces inside the ball. The obtained lower bound is not sharp. We provide computational experiments and state a conjecture about an improved index lower bound. Independently of it we also find an upper bound on the index of Fraser-Sargent surfaces inside the ball.