Harmonic metrics of $\mathrm{SO}_{0}(n,n)$-Higgs bundles in the Hitchin section on non-compact hyperbolic surfaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912438971006976 |
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| author | Ma, Weihan |
| author_facet | Ma, Weihan |
| contents | Let $X$ be a Riemann surface. Hitchin constructed the $G$-Higgs bundles in the Hitchin section for a split real form $G$ of a complex simple Lie group,using the canonical line bundle $K$ and some holomorphic differentials $\boldsymbol{q}$. We study the case of ${\mathrm{SO}_0(n,n)}$. In our work, we establish the existence of harmonic metrics for these Higgs bundles, which are compatible with the ${\mathrm{SO}_0(n,n)}$-structure on any non-compact hyperbolic Riemann surface. Furthermore, these harmonic metrics weakly dominate $h_X$, the natural diagonal harmonic metric induced by the unique complete Kähler hyperbolic metric $g_X$ on $X$. Assuming these holomorphic differentials are all bounded with respect to $g_X$, we prove the uniqueness of such a harmonic metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic metrics of $\mathrm{SO}_{0}(n,n)$-Higgs bundles in the Hitchin section on non-compact hyperbolic surfaces Ma, Weihan Differential Geometry 53C07, 58E15, 14D21, 81T13 Let $X$ be a Riemann surface. Hitchin constructed the $G$-Higgs bundles in the Hitchin section for a split real form $G$ of a complex simple Lie group,using the canonical line bundle $K$ and some holomorphic differentials $\boldsymbol{q}$. We study the case of ${\mathrm{SO}_0(n,n)}$. In our work, we establish the existence of harmonic metrics for these Higgs bundles, which are compatible with the ${\mathrm{SO}_0(n,n)}$-structure on any non-compact hyperbolic Riemann surface. Furthermore, these harmonic metrics weakly dominate $h_X$, the natural diagonal harmonic metric induced by the unique complete Kähler hyperbolic metric $g_X$ on $X$. Assuming these holomorphic differentials are all bounded with respect to $g_X$, we prove the uniqueness of such a harmonic metric. |
| title | Harmonic metrics of $\mathrm{SO}_{0}(n,n)$-Higgs bundles in the Hitchin section on non-compact hyperbolic surfaces |
| topic | Differential Geometry 53C07, 58E15, 14D21, 81T13 |
| url | https://arxiv.org/abs/2408.15278 |