Large-scale rank and rigidity of the Weil-Petersson metric
This paper establishes several rigidity theorems regarding the coarse geometry of the Teichmüller space with Weil-Peterson metric, which the author calls the Weil-Peterson space. To do so, he construct a coarse median structure of the Weil-Peterson space, which in particular gives the coarse rank of the Weil-Peterson space originally proven by \textit{A. Eskin} et al. [Duke Math. J. 166, No. 8, 1517--1572 (2017; Zbl 1373.32012)]. Royden's theorem asserts that every isometry of the Teichmüller space with Teichmüller metric is given by a mapping class of the surface, and Masur and Wolf proved Royden's theorem with the Weil-Peterson metric. The author proves a coase version of Royden's theorem for the Weil-Peterson space, except when the complexity of the surface is low. The author also investigates the asymptotic cone of the Weil-Peterson space, and shows that it is uniquely determined by the base surface, similarly, for most of the cases.
scientific article; zbMATH DE number 7227240
scientific article