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

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 \(g \le 6\), then the multiplicity of the lowest eigenvalue \(\lambda _1 = -2\) is 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.