Ramification and Finiteness of Equivariant Minimal Surfaces

Abstract

Let $S$ be a closed oriented surface of genus $g$, with $g\ \geq \ 2$, and let $\mathcal X_0^{\mathrm{good}}$ be the good locus of $\operatorname{PU}(2,\,1)$ representations with Toledo invariant zero. For $1\ \leq \ b\ \leq\ 2g-3$, we show that representations admitting an equivariant minimal map of total ramification at least $b$, for some marked complex structure, form a countable union of compact null sets of Hausdorff dimension at most $16g-16-2b$. For larger $b$, this locus is empty. Consequently, on a dense $G_\delta$ subset of full measure in $\mathcal X_0^{\mathrm{good}}$, every nonconstant equivariant minimal map is an immersion, for every marked complex structure. On the convex cocompact locus, a dense $G_\delta$ subset of full measure consists of representations admitting a finite nonempty set of marked equivariant minimal immersions.