Computation of harmonic functions on higher genus surfaces

Abstract

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a harmonic function in a subdomain of a compact Riemann surface may be approached by globally defined harmonic functions with prescribed poles. We illustrate the effectiveness and the impact of the method solving general boundary value Laplace problems in subdomains of the surface; we lay the groundwork for this numerical method in Riemann surfaces represented by a gluing of hyperbolic polygons. In particular, we give a general approximation procedure that computes this basis efficiently with arbitrary precision.