Index and nullity of minimal surface doublings, I

TL;DR

This paper analyzes the index and nullity of a family of minimal surface doublings in the three-sphere, establishing their stability properties and uniqueness for large genus values.

Contribution

It provides explicit calculations of index and nullity for high-genus minimal surface doublings, showing they are $C^1$-isolated with no exceptional Jacobi fields.

Findings

Index is $2m+7$ for genus $m+1$ surfaces.

Nullity is consistently 6.

Surfaces are $C^1$-isolated with no exceptional Jacobi fields.

Abstract

We prove that for any large enough $m \in\mathbb{N}$, the genus $\gamma=m+1$ equator-poles minimal surface doubling of the equatorial two-sphere $\Sigma^0 = \mathbb{S}^2_{\mathrm{eq}}$ in the round three-sphere $\mathbb{S}^3$, which has two catenoidal bridges at the poles and $m$ bridges equidistributed along the equatorial circle $\mathscr{C}$ of $\Sigma^0 $ and was discovered in earlier work of Kapouleas, has index $2\gamma+5=2m+7$ and nullity $6$, and so it has no exceptional Jacobi fields and is $C^1$-isolated.