Weil--Petersson homeomorphisms, minimal lagrangian diffeomorphisms, and maximal surfaces in anti-de Sitter space

In this paper, we study the class of Weil–Petersson circle homeomorphisms from the point of view of three-dimensional anti-de Sitter space ${\mathbf{A}\mathbf{d}\mathbf{S}}^{2,1}$. We show that a homeomorphism $\phi :{\mathbf{R}\mathbf{P}}^1\to {\mathbf{R}\mathbf{P}}^1$ is Weil–Petersson if and only if its graph, viewed as a curve in the boundary at infinity of ${\mathbf{A}\mathbf{d}\mathbf{S}}^{2,1}$, is the asymptotic boundary of a complete maximal spacelike surface in ${\mathbf{A}\mathbf{d}\mathbf{S}}^{2,1}$ with finite renormalized area. As an application, we obtain the following AdS-independent result in Teichmüller theory: a homeomorphism is Weil–Petersson if and only if its minimal lagrangian extension to ${\mathbf{H}}^2$ has square-integrable Beltrami differential. We also provide two further new technical characterizations, which we believe to be of independent interest, and which are essential for the proofs of our main results.