Comparing invariant distances and conformal metrics on Riemann surfaces (Q5935809)

Various types of distances or metrics are defined on a Riemann surface \(X\) in terms of bounded or positive harmonic functions. The authors prove sharp inequalities comparing the hyperbolic distance \(d_X\) (or the hyperbolic metric) to these distances (metrics). In case the surface is simply connected, it is proved that the above inequalities are identities. For the non-simply connected case, the cases of equality are precisely determined. A typical result of the paper is the following: Let \(b_X(x,y)=\sup |u(x)-u(y)|\), the supremum being taken over all harmonic functions on \(X\) with \(0<u<1\). Then \[ b_X(x,y)\leq \tfrac{8}{\pi} \arctan (\tanh (d_X(x,y)/2)). \] If \(X\) is not simply connected and equality holds, then \(X\) is (up to conformal equivalence) an annulus and \(x,y\) are symmetric about the circle of symmetry.