Minimal tori with low nullity

TL;DR

This paper investigates minimal tori in the 3-sphere with low natural nullity, constructing examples, analyzing their symmetries, and relating known minimal tori to their nullity properties.

Contribution

It constructs all minimal tori in S^3 with natural nullity less than 8 and relates these to known examples, providing new insights into their symmetry and nullity characteristics.

Findings

Lawson and Hsiang examples have natural nullity equal to their Killing nullity

Minimal tori with natural nullity ≤6 have non-trivial isometry groups

Constructs minimal immersions of R^2 in S^3 encompassing all low nullity tori

Abstract

The nullity of a minimal submanifold $M\subset S^{n}$ is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions $SO(n+1)$ of the ambient space, modulo those motions which preserve $M$, whose dimension is the Killing nullity $kn(M)$ of $M$. In the case of 2-dimensional tori $M$ in $S^{3}$, there is an additional naturally-defined 2-dimensional subspace; the dimension of the sum of the action of the rigid motions and this space is the natural nullity $nnt(M)$. In this paper we will study minimal tori in $S^{3}$ with natural nullity less than 8. We construct minimal immersions of the plane $R^{2}$ in $S^{3}$ that contain all possible examples of tori with $nnt(M)<8$. We prove that the examples of Lawson and Hsiang with $kn(M)=5$ also have $nnt(M)=5$, and we prove that if the $nnt(M)\le6$ then the group of isometries of $M$ is not trivial.