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

Abstract

In this paper, we study the class of Weil--Petersson circle homeomorphisms from the point of view of three-dimensional anti-de Sitter space $\mathbf{AdS}^{2,1}$. We show that a homeomorphism $\varphi:\mathbf{RP}^1\to\mathbf{RP}^1$ is Weil--Petersson if and only if its graph, viewed as a curve in the boundary at infinity of $\mathbf{AdS}^{2,1}$, is the asymptotic boundary of a complete maximal spacelike surface in $\mathbf{AdS}^{2,1}$ with finite renormalized area. As an application, we obtain the following AdS-independent result in Teichm\"uller 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.