WEIL-PETERSSON TEICHMÜLLER SPACE

The paper presents some recent results on the Weil-Petersson geometry theory of the universal Teichmüller space, a topic which is important in Teichmüller theory and has wide applications to various areas such as mathematical physics, differential equation and computer vision. (1) It is shown that a sense-preserving homeomorphism ℎ on the unit circle belongs to the Weil-Petersson class, namely, ℎ can be extended to a quasiconformal mapping to the unit disk whose Beltrami coefficient is square integrable in the Poincaré metric if and only if ℎ is absolutely continuous and logℎʹ belongs to the Sobolev class H 1 2 . This solves an open problem posed by Takhtajan-Teo in 2006 and investigated later by Figalli, Gay-Balmaz-Marsden-Ratiu and others. The intrinsic characterization (1) of the Weil-Petersson class has the following applications which are also explored in this paper: (2) It is proved that there exists a quasisymmetric homeomorphism of the Weil-Petersson class which belongs neither to the Sobolev class H 3 2 nor to the Lipschitz class Λ¹, which was conjectured very recently by Gay-Balmaz-Ratiu when studying the classical Euler-Poincaré equation in the new setting that the involved sense-preserving homeomorphisms on the unit circle belong to the Weil-Petersson class. (3) It is proved that the flows of the H 3 2 vector fields on the unit circle are contained in the Weil-Petersson class, which was also conjectured by Gay-Balmaz-Ratiu in their above mentioned research. (4) A new metric is introduced on the Weil-Petersson Teichmüller space. It is shown to be topologically equivalent to the Weil-Petersson metric.