The Second Variation of Holomorphic Curves in the 6-Sphere

Abstract: The 6-sphere is the only $n$-sphere with $n > 2$ that admits an almost-complex structure. Equipping the round 6-sphere with its standard ($G_2$-invariant) almost-complex structure, the holomorphic curves in $S^6$ are minimal surfaces, and play an important role in $G_2$-geometry. These surfaces exist in abundance: by a remarkable theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in $S^6$ as a holomorphic curve of "null-torsion."

While holomorphic curves in $S^6$ are area-minimizing to first order, they are not area-minimizing to second order. This failure is encoded by the spectrum of the Jacobi operator, which contains information such as the Morse index and nullity. For closed, null-torsion holomorphic curves of low genus, we explicitly compute the multiplicity of the first Jacobi eigenvalue. Moreover, for all genera, we give a simple lower bound for the nullity in terms of the area and genus. Time permitting, we will also outline some recent results in the setting of holomorphic curves with boundary.

differential geometry

Audience: researchers in the topic

( paper | video )

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Virtual seminar on geometry with symmetries

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