Holomorphic curves in the 6-pseudosphere and cyclic surfaces

2023/02/22 by B. Collier, Collier, Brian, Jérémy Toulisse +1 · 3 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2302.11516

Abstract

The space H^4,2 of vectors of norm -1 in 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 G₂'. In this paper we consider a class of holomorphic curves in 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 are naturally parameterized by certain G₂'-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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• Cyclic Higgs bundles and minimal surfaces in pseudo-hyperbolic spaces
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