  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.
  \noindent (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
squarely integrable in the Poincar\'e metric if and only if $h$ is absolutely
continuous such that $\log h'$ belongs to the Sobolev class $H^{\frac 12}$.
This solves an open problem posed by Takhtajan-Teo [TT2] in 2006 and
investigated later by Figalli [Fi], Gay-Balmaz-Marsden-Ratiu ([GMR], [GR]) and
others.
  \noindent The intrinsic characterization (1) of the Weil-Petersson class has
the following applications which are also explored in this paper:
  \noindent (2) It is proved that there exists a quasisymmetric homeomorphism
of the Weil-Petersson class which belongs neither to the Sobolev class
$H^{\frac 32}$ nor to the Lipschitz class $\Lambda^1$, which was conjectured
very recently by Gay-Balmaz-Ratiu [GR] 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.
  \noindent (3) It is proved that the flows of the $H^{\frac 32}$ vector fields
on the unit circle are contained in the Weil-Petersson class, which was also
conjectured by Gay-Balmaz-Ratiu [GR] during their above mentioned research.
  \noindent (4) A new metric is introduced on the Weil-Petersson Teichm\"uller
space and is shown to be topologically equivalent to the Weil-Petersson metric.
