Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912161196933120 |
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| author | Dai, Song Li, Qiongling |
| author_facet | Dai, Song Li, Qiongling |
| contents | A rank $n$ Higgs bundle $(E,θ)$ is called generically regular nilpotent if $θ^n=0$ but $θ^{n-1}\neq 0$. 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 $\mathbb{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces Dai, Song Li, Qiongling Differential Geometry A rank $n$ Higgs bundle $(E,θ)$ is called generically regular nilpotent if $θ^n=0$ but $θ^{n-1}\neq 0$. 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 $\mathbb{D}$. |
| title | Harmonic metrics of generically regular nilpotent Higgs bundles over non-compact surfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2412.14429 |