Complete finite energy equivariant minimal surfaces in CH^2

Abstract

We will discuss $\rho$-equivariant conformal minimal immersions of finite energy into $\mathbb{CH}^2$. We prove that the induced metric is complete at a puncture $p$ if and only if the holonomy of $\rho$ around $p$ is parabolic. In this case $f$ is proper and $\partial f$ has pole at each puncture, so on the compactification $\Sigma$ we obtain a parabolic $\mathrm{PU}(2,1)$–Higgs bundle $(E,\Phi)$ with nilpotent residues and zero parabolic weights. This yields an Osserman-type extension principle for minimal surfaces in $\mathbb{CH}^2$ with complete ends. We further provide an algebro–geometric characterization of these immersions in terms of their associated parabolic Higgs data. This is a joint work with Indranil Biswas and John Loftin.