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

Given a noncompact Riemann surface ${\mathrm{\Sigma }}_0=\mathrm{\Sigma }\setminus P$, where $P$ is a finite subset of a compact connected Riemann surface $\mathrm{\Sigma }$, and a reductive representation $\rho :{\pi }_1({\mathrm{\Sigma }}_0)⟶\mathrm{P}\mathrm{U}(2,1)$, 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 $\mathrm{P}\mathrm{U}(2,1)$–Higgs bundles with nilpotent residues and satisfying concrete parabolic slope inequalities. We also discuss complete ends and construct explicit families of $\rho$ equivariant proper ${\mathbb{C}\mathbb{H}}^2$ $n$–noids on ${\mathbb{C}\mathbb{P}}^1\setminus P$ for $\mathrm{\mid }P\mathrm{\mid }\ge 5$.