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

Abstract

Given a noncompact Riemann surface $\hat{I}{\pounds }_0=\hat{I}\pounds \setminus P$, where $P$ is a finite subset of a compact connected Riemann surface $\hat{I}\pounds$, and a reductive representation $\ddot{I}:\ddot{I}{€}_1(\hat{I}{\pounds }_0)⟶\mathrm{PU}(2,1)$, we prove that any finite--energy $\ddot{I}$--equivariant conformal minimal immersion is proper around every cusp if and only if the peripheral holonomy of $\ddot{I}$ 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 $\ddot{I}$ equivariant proper ${\mathbb{CH}}^2$ $n$--noids on ${\mathbb{CP}}^1\setminus P$ for $|P|\ge 5$.

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