Equivariant finite energy proper minimal surfaces in $\mathbb{CH}^2$

TL;DR

This paper studies equivariant minimal surfaces in complex hyperbolic space, characterizing properness via holonomy, and constructs explicit examples of such surfaces with prescribed properties.

Contribution

It provides a complete characterization of properness for finite-energy equivariant minimal surfaces in $ ext{CH}^2$ based on peripheral holonomy and offers explicit parametrizations and examples.

Findings

Properness linked to parabolic peripheral holonomy.

Explicit parametrization of complete finite-energy immersions.

Construction of explicit $ ho$-equivariant $ ext{CH}^2$ $n$-noids.

Abstract

Given a noncompact Riemann surface $\Sigma_0\,=\, \Sigma \setminus P$, where $P$ is a finite subset of a compact connected Riemann surface $\Sigma$, and a reductive representation $\rho\,:\,\pi_1(\Sigma_0)\,\longrightarrow\, \mathrm{PU}(2,1)$, we prove that any finite--energy $\rho$--equivariant conformal minimal immersion is proper around every cusp if and only if the peripheral holonomy of $\rho$ is parabolic. Assuming parabolic peripheral holonomy, we give an explicit parametrization of complete finite--energy immersions in the mixed case in terms of tame parabolic $\mathrm{PU}(2,1)$--Higgs bundles with nilpotent residues and satisfying concrete parabolic slope inequalities. We also discuss complete ends and construct explicit families of $\rho$ equivariant proper $\mathbb{CH}^2$ $n$--noids on $\mathbb{CP}^1\setminus P$ for $|P|\,\ge\, 5$.