Minimal surfaces are area-minimizing to first order, but not necessarily to second-order. The extent to which a minimal surface is (or isn’t) area-minimizing to second-order is encoded in its Jacobi operator. However, for a given minimal surface, computing the spectrum of its Jacobi operator — its eigenvalues and their multiplicities — is generally a non-trivial task, even when the ambient space is .

In this talk, we will discuss a class of minimal surfaces in the round 6-sphere known as “null-torsion holomorphic curves.” These surfaces are of interest to G2 geometry, and exist in abundance. Indeed, by a remarkable theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded as a null-torsion holomorphic curve in .

For null-torsion holomorphic curves of low genus, we will compute the multiplicity of the first Jacobi eigenvalue. Moreover, for all genera, we will give a simple lower bound for the nullity (the multiplicity of the zero eigenspace) in terms of the area and genus. We expect that these results will have implications for the deformation theory of asymptotically conical associative 3-folds in euclidean 7- space.