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

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.