Minimal harmonic functions on a Riemann surface
Yukimasa Sumita · Kodai Mathematical Journal · 1966
The aim of the present paper is to give another proof of the well known fact that if there exists an fflλminimal or iiffi-minimal function on a Riemann surface then there exists no non-constant analytic function with a finite Dirichlet integral or no non-constant bounded analytic function respectively.This fact was proved by several authors by using the universal covering surfaces or the compactification of the original Riemann surfaces;In the present paper we prove this without using the universal covesing surface or the compactification.We denote by S an arbitrary Riemann surface and by G an arbitrary open set with the relative boundary dG consisting of at most enumerable number of piecewise analytic Jordan curves which does not cluster on S. It is sufficient for our purpose to consider only such an open set G satisfying this condition.Let us denote by HP 0 (G) the class of all non-negative harmonic functions defined on G which vanish on dG.If UGHP 0 (G), we define u* on S by «*=« on G and u*=0 on S-G.We denote by E G u for u£HP 0 {G) the extremisation of u over G and by I G v the inextremisation of v for v € HP(S), where HP(S) is the class of all non-negative harmonic functions on S. If there occurs no confusion, we write merely Eu, Iv instead of EGU, I G V.Eu and Iv are defined by (1)Eu=inί ιv[w^u on G, w e HP(S)],(2) Iv=snpw[w^v on G, WGHP 0 (G)].