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

TL;DR

This paper studies harmonic metrics on a special class of nilpotent Higgs bundles over non-compact surfaces, establishing uniqueness and maximality results, and applies these findings to minimal surfaces in hyperbolic space.

Contribution

It extends the theory of harmonic metrics to generically regular nilpotent Higgs bundles on non-compact surfaces and generalizes a key theorem for prescribed curvature equations.

Findings

Unique maximal harmonic metric exists for such Higgs bundles.

Generalization of Kalka-Yang's theorem to non-compact hyperbolic surfaces.

Branched sets of minimal disks in hyperbolic space relate to holomorphic self-maps.

Abstract

A rank $n$ Higgs bundle $(E,\theta)$ is called generically regular nilpotent if $\theta^n=0$ but $\theta^{n-1}\neq 0$. 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 $\mathbb{D}$.