A homeomorphism of the circle is Weil-Petersson precisely when its graph bounds a maximal surface in AdS^{2,1} with finite renormalized area.
Takhtajan and Lee-Peng Teo, Weil-Petersson metric on the universal Teichmüller space , Memoirs of the American Mathematical Society 183 (2006), no
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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.