Sario’s Potentials and Analytic Mappings

Abstract

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

Locations

• Nagoya Mathematical Journal - View - PDF
• Project Euclid (Cornell University) - View - PDF