Holomorphic curves in the 6-pseudosphere and cyclic surfaces

TL;DR

This paper studies a special class of holomorphic curves called alternating in the 6-pseudosphere, linking their geometry to $G_2'$-Higgs bundles and revealing their rigidity and moduli space structure.

Contribution

It introduces alternating holomorphic curves in $ extbf{H}^{4,2}$, establishes a Frenet framing, and connects these curves to $G_2'$-Higgs bundles and the moduli space structure.

Findings

Alternating holomorphic curves admit a Frenet framing.

The moduli space of equivariant alternating curves is a fibration over Teichmüller space.

Such curves are infinitesimally rigid.

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 $G_2'$. 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 are naturally parameterized by certain $G_2'$-Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichm\"uller 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.