A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials

In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.