Ramification and Finiteness of Equivariant Minimal Surfaces

Let $S$ be a closed oriented surface of genus $g$, with $g\ge 2$, and let ${\mathcal{X}}_0^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$ be the good locus of $\mathrm{PU}(2,1)$ representations with Toledo invariant zero. For $1\le b\le 2g-3$, 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 ${\mathcal{X}}_0^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{d}}$, 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.