Maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}{\mathbb{S}}^{2,1}$

Abstract

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 $T^{\ast }{T}_0(1)$ of Weil-Petersson universal Teichmüller space $T_0(1)$. We prove that the Mess map defines a symplectic diffeomorphism from $T^{\ast }{T}_0(1)$ to $T_0(1)\times T_0(1)$, with respect to the canonical symplectic form on $T^{\ast }{T}_0(1)$ 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 $T_0(1{)}^{\pm }\subset T^{\ast }{T}_0(1)$, which bijectively parametrize the space of maximal discs of Weil-Petersson class in $\mathbb{A}\mathrm{d}{\mathbb{S}}^{2,1}$.

31 pages