Equivariant minimal surfaces in $\mathbb{CH}^2$ and their Higgs bundles

TL;DR

This paper classifies equivariant minimal surfaces in complex hyperbolic space via Higgs bundles, revealing the structure of their moduli space and linking geometric properties to algebraic invariants.

Contribution

It provides a complete description of the Higgs bundle structure for all equivariant minimal immersions into $ ext{CH}^2$, and characterizes the moduli space of such pairs.

Findings

Moduli space decomposes into finitely many complex manifolds.

Holomorphic and anti-holomorphic maps form distinct components.

Non-holomorphic maps' components match the dimension of the representation variety.

Abstract

This paper gives a construction for all minimal immersions $f$ of the Poincar\'{e} disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $\rho$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space of pairs $(\rho,f)$ is a disjoint union of finitely many complex manifolds, whose structure we fully describe. The holomorphic (or anti-holomorphic) maps provide multiple components of this union, as do the non-holomorphic maps. Each of the latter components has the same dimension as the representation variety for $PU(2,1)$, and is indexed by the number of complex and anti-complex points of the immersion. These numbers determine the Toledo invariant and the Euler number of the normal bundle of the immersion. We show that there is an open set of quasi-Fuchsian representations of Toledo invariant zero for which the minimal surface is unique and Lagrangian.