Holomorphic curves in the 6-pseudosphere and cyclic surfaces

Collier, Brian; Toulisse, Jérémy

doi:10.1090/tran/9172

Holomorphic curves in the $6$-pseudosphere and cyclic surfaces
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by Brian Collier and Jérémy Toulisse;

Trans. Amer. Math. Soc. 377 (2024), 6465-6514

DOI: https://doi.org/10.1090/tran/9172

Published electronically: June 21, 2024

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Abstract:

The space $\mathbf {H}^{4,2}$ of vectors of norm $-1$ in $\mathbb {R}^{4,3}$ has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form $\mathsf {G}_2’$. In this paper we consider a class of holomorphic curves in $\mathbf {H}^{4,2}$ which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group is naturally parameterized by certain $\mathsf {G}_2’$-Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichmüller space with a holomorphic action of the mapping class group. Using a generalization of Labourie’s cyclic surfaces, we then show that equivariant alternating holomorphic curves are infinitesimally rigid.

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