The Jacobi Spectrum of Null-Torsion Holomorphic Curves in the 6-Sphere

Jesse Madnick

15-Jul-2021, 23:00-00:00 (3 years ago)

Abstract: 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 by its Jacobi operator. However, for a given minimal surface, computing the spectrum of the Jacobi operator — i.e., the eigenvalues and their multiplicities — is generally a non-trivial task. \indent 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 $G_2$ 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 $S^6$. \indent 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 $R^7$.

differential geometry

Audience: researchers in the topic

( video )


The 2nd Geometric Analysis Festival

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