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arXiv · 2412.14429

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

Song Dai·Qiongling Li
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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 $\mathbb{D}$.

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Song Dai, Qiongling Li. 2024-12-19. Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces. https://arxiv.org/abs/2412.14429

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