A homeomorphism of the circle is Weil-Petersson precisely when its graph bounds a maximal surface in AdS^{2,1} with finite renormalized area.
Schoen, The role of harmonic mappings in rigidity and deformation problems, Complex Geometry (Osaka, 1990), Lecture Notes in Pure and Appl
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Weil--Petersson homeomorphisms, minimal lagrangian diffeomorphisms, and maximal surfaces in anti-de Sitter space
A homeomorphism of the circle is Weil-Petersson precisely when its graph bounds a maximal surface in AdS^{2,1} with finite renormalized area.