Holomorphic curves in the 6-pseudosphere and cyclic surfaces

The space H 4, 2 H^4, 2 of vectors of norm − 1 -1 in R 4, 3 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 ′ G₂’. In this paper we consider a class of holomorphic curves in H 4, 2 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 ′ 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.