Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909018420674560 |
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| author | Park, Jinsung |
| author_facet | Park, Jinsung |
| contents | We introduce maximal discs of Weil-Petersson class in the 3-dimensional Anti-de Sitter space $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$, whose parametrization space can be identified with the cotangent bundle $T^*T_0(1)$ of Weil-Petersson universal Teichmüller space $T_0(1)$. We prove that the Mess map defines a symplectic diffeomorphism from $T^*T_0(1)$ to $T_0(1)\times T_0(1)$, with respect to the canonical symplectic form on $T^*T_0(1)$ and the difference of pullbacks of the Weil-Petersson symplectic forms from each factor of $T_0(1)\times T_0(1)$. Furthermore, we show that the functional given by the anti-holomorphic energies of the induced Gauss maps associated with maximal discs of Weil-Petersson class serves as a Kähler potential for the restriction of the canonical symplectic form to certain submanifolds $T_0(1)^\pm \subset T^*T_0(1)$, which bijectively parametrize the space of maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06498 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$ Park, Jinsung Symplectic Geometry Mathematical Physics Differential Geometry Geometric Topology 30F60, 32G15, 53C50, 53A10 We introduce maximal discs of Weil-Petersson class in the 3-dimensional Anti-de Sitter space $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$, whose parametrization space can be identified with the cotangent bundle $T^*T_0(1)$ of Weil-Petersson universal Teichmüller space $T_0(1)$. We prove that the Mess map defines a symplectic diffeomorphism from $T^*T_0(1)$ to $T_0(1)\times T_0(1)$, with respect to the canonical symplectic form on $T^*T_0(1)$ and the difference of pullbacks of the Weil-Petersson symplectic forms from each factor of $T_0(1)\times T_0(1)$. Furthermore, we show that the functional given by the anti-holomorphic energies of the induced Gauss maps associated with maximal discs of Weil-Petersson class serves as a Kähler potential for the restriction of the canonical symplectic form to certain submanifolds $T_0(1)^\pm \subset T^*T_0(1)$, which bijectively parametrize the space of maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$. |
| title | Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$ |
| topic | Symplectic Geometry Mathematical Physics Differential Geometry Geometric Topology 30F60, 32G15, 53C50, 53A10 |
| url | https://arxiv.org/abs/2412.06498 |