In this paper, we consider conformal invariants defined by various spaces of harmonic functions on Riemann surfaces. The Harnack distance is a typical one. We give sharp inequalities comparing those invariants with the hyperbolic metric on the Riemann surface and we determine when equalities hold. We also describe the Harnack distance in terms of the Martin compactification and discuss some properties of the distance.
"Conformal invariants defined by harmonic functions on Riemann surfaces." J. Math. Soc. Japan 68 (1) 441 - 458, January, 2016. https://doi.org/10.2969/jmsj/06810441