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  1. ; ; ; ; (, Annals of Mathematics)
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  2. ; (, Transactions of the American Mathematical Society)
    The space H 4 , 2 \mathbf {H}^{4,2} of vectors of norm −<#comment/> 1 -1 in R 4 , 3 \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 G 2 ′ \mathsf {G}_2’ . In this paper we consider a class of holomorphic curves in H 4 , 2 \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 G 2 ′ \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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