Аннотация:
Let $\Gamma$ be a Fuchsian group acting on the upper half-plane $\mathbb{H}$. The square integrable Teichmüller space is a subset of the Teichmüller space $T(\Gamma)$ which consists of the Teichmüller equivalence classes with $p$ -integrable Beltrami coefficients as their representatives. Here, a Beltrami coefficient is square integrable if it is square integrable with respect to the hyperbolic metric on the Riemann surface $\mathbb{H}/\Gamma$.
When $\Gamma$ is of analytically finite type, $T(\Gamma)$ has a finite dimensional Hilbert manifold structure. The Weil-Petersson metric is an Hermitian metric on this structure. It is known that this metric is Kähler and is not complete. Moreover, it has the negative holomorphic sectional curvature, the negative Ricci curvature, and the negative Scalar curvature. In this talk, we will extend the concept of the Weil-Petersson metric to the square integrable Teichmüller space of Fuchsian groups of analytically infinite type.
