The family of (single-valued) analytic functions with finite Dirichlet integrals on a Riemann surface R will be denoted by AD(R). We also use the notation AD without reference to the surface R and we let O AD signify the class of Riemann surfaces for which AD does not contain any non-constant functions. As a consequence of the maximum modulus principle every closed Riemann surface is in O AD . We are therefore mainly interested in open Riemann surfaces which belong to this class.


Riemann Surface Covering Surface Boundary Component Annular Region Ideal Boundary 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1970

Authors and Affiliations

  • L. Sario
    • 1
  • M. Nakai
    • 2
  1. 1.Unversity of CaliforniaCaliforniaUSA
  2. 2.Nagoya UnversityGermany

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