Index of bipolar surfaces to Otsuki tori

TL;DR

This paper investigates the Morse index and nullity of bipolar surfaces to Otsuki tori in spheres, providing bounds and conjectures for cases near a specific rational parameter, advancing understanding of minimal tori in higher-dimensional spheres.

Contribution

It offers new bounds on the Morse index and nullity of bipolar Otsuki tori and proposes a conjecture for the general case based on numerical evidence.

Findings

Bounds on Morse index and nullity near p/q close to √2/2

Numerical evidence supporting a conjecture for the general case

Enhanced understanding of minimal tori in spheres

Abstract

For each rational number $p/q\in (1/2,\sqrt 2/2)$ one can construct an $\mathbb S^1$-equivariant minimal torus in $\mathbb S^3$ called Otsuki torus and denoted by $O_{p/q}$. The Lawson's bipolar surface construction applied to $O_{p/q}$ gives a minimal torus $\widetilde O_{p/q}$ in $\mathbb S^4$. In this paper we give upper and lower bounds on the Morse index and the nullity of these tori for $p/q$ close to $\sqrt 2/2$. We also state a numerically assisted conjecture concerning the general case.