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

DOI10.1007/S12220-022-01040-9arXiv2101.09580MaRDI QIDQ6358874

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Publication date: 23 January 2021

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 $4pid$, focusing on the spectrum of the Jacobi operator. We show that if $gleq6$, then the multiplicity of the lowest eigenvalue $lambd{a}_1=-2$ 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.

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