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 $T^*T_0(1)$ of Weil-Petersson universal Teichm\"uller 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 K\"ahler 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 $\mathbb{A}\mathrm{d}\mathbb{S}^{2,1}$.