Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces

Abstract

A rank $n$ Higgs bundle $(E,\theta )$ is called generically regular nilpotent if ${\theta }^n=0$ but ${\theta }^{n-1}\ne 0$. We show that for a generically regular nilpotent Higgs bundle, if it admits a harmonic metric, then its graded Higgs bundle admits a unique maximal harmonic metric. The proof relies on a generalization of Kalka-Yang's theorem for prescribed curvature equation over a non-compact hyperbolic surface to a coupled system. As an application, we show that the branched set of a branched minimal disk in ${\mathbb{H}}^3$ has to be the critical set of some holomorphic self-map of $\mathbb{D}$.

23 pages, comments are very welcome