The paper presents some recent results on the Weil-Petersson geometry theory of the universal Teichm\"uller space, a topic which is important in Teichm\"uller 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 Poincar\'e metric if and only if $h$ is absolutely continuous and $\log h'$ belongs to the Sobolev class $H^{1\over2}$. 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 $H^{3\over2}$ nor to the Lipschitz class $\Lambda^1$, which was conjectured very recently by Gay-Balmaz-Ratiu when studying the classical Euler-Poincar\'e 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 $H^{3\over2}$ 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 Teichm\"uller space. It is shown to be topologically equivalent to the Weil-Petersson metric.


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pp. 1041-1074
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