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

2024/12/19 by Song Dai, Dai, Song, Qiongling Li +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2412.14429

Abstract

A rank n Higgs bundle (E,θ) is called generically regular nilpotent if θⁿ=0 but θ^n-1≠ 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 ℍ³ has to be the critical set of some holomorphic self-map of \mathbbD.

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