An estimate of the Bergman distance on Riemann surfaces

Bo-Yong Chen, Yuanpu Xiong

Let $M$ be a hyperbolic Riemann surface with the first eigenvalue ${\lambda }_1(M)>0$. Let $\rho$ denote the distance from a fixed point $x_0\in M$ and $r_x$ the injectivity radius at $x$. We…