The Second Variation for Null-Torsion Holomorphic Curves in the 6-Sphere

TL;DR

This paper investigates the second variation of area for null-torsion holomorphic curves in the 6-sphere, revealing spectral properties of the Jacobi operator that depend on genus and area, with implications for geometric deformation theory.

Contribution

It provides explicit spectral results for the Jacobi operator on null-torsion holomorphic curves, extending understanding of their stability and deformation characteristics.

Findings

Lowest eigenvalue multiplicity equals 4d for g ≤ 6

Nullity is at least 2d + 2 - 2g for all genus

Results have implications for deformation theory of associative 3-folds

Abstract

In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces. This class of minimal surfaces is quite rich: By a theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in the round 6-sphere as a null-torsion holomorphic curve. In this work, we study the second variation of area for compact null-torsion holomorphic curves $\Sigma$ of genus $g$ and area $4\pi d$, focusing on the spectrum of the Jacobi operator. We show that if $g \leq 6$, then the multiplicity of the lowest eigenvalue $\lambda_1 = -2$ is exactly equal to $4d$. Moreover, for any genus, we show that the nullity is at least $2d + 2 - 2g$. These results are likely to have implications for the deformation theory of asymptotically conical associative $3$-folds in $\mathbb{R}^7$, as studied by Lotay.