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In the round 6-sphere, null-torsion holomorphic curves are fundamentalexamples of minimal surfaces. This class of minimal surfaces is quite rich: Bya theorem of Bryant, extended by Rowland, every closed Riemann surface may beconformally 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-torsionholomorphic curves $\Sigma$ of genus $g$ and area $4\pi d$, focusing on thespectrum of the Jacobi operator. We show that if $g \leq 6$, then themultiplicity 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 theoryof asymptotically conical associative $3$-folds in $\mathbb{R}^7$, as studiedby Lotay.

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