Mitsuru Nakai

A measure on the harmonic boundary of a Riemann surface.

In the usual theory of harmonic functions on a plane domain, the fact that the boundary of the domain is realized relative to the complex plane plays an essential role and supplies many powerful tools, for instance, the solution of Dirichlet problem. But in the theory of harmonic functions on a general domain, i.e. on a Riemann surface, the main difficulty arises from the lack of the “visual” boundary of the surface. Needless to say, in general we cannot expect to get the “relative” boundary with respect to some other larger surface. In view of this, we need some “abstract” compactifications. It seems likely that we cannot expect to get the “universal” boundary which is appropriate for any harmonic functions since there exist many surfaces which do not admit some classes of harmonic functions as the classification theory shows. Hence we need many compactifications corresponding to what class of harmonic functions we are going to investigate.

The Space of Dirichlet-Finite Solutions of the Equation Δu = Pu on a Riemann Surface

respectively. By a solution of (1) on R we mean a twice continuously differentiable function which satisfies the relation (l) on R. We denote by PB (or PD or PE) the totality of bounded (or Dirichlet-finite or energy-finite) solutions of (1) on R. We also denote by PBD ^PBΠPD and PBE ^PBf PE. If the class X contains no non-constant function, then we denote the fact by R e Ox, where X stands for one of classes PB, PD, PE, PBD or PBE, Here we remaik that a constant solution of (1) is necessarily zero, since we have assumed that P

Circle means of Green's functions

0 on £ We also use the notation R^OG to denote the fact that R is a

Dirichlet finite biharmonic functions

Consider the polar coordinate differentials (dr, dθ) on a hyperbolic Riemann surface R with center z 0 ∈ R which are given by where G R ( z , ζ) is the Green’s function on R with pole ζ ∈ R.

On evans potential

From the class A of analytic functions we proceed to the class H of harmonic functions. The latter are in a sense more flexible than the former and thus easier to treat. In particular the solvability of the Dirichlet problem makes it possible to obtain detailed information on the causes of degeneracy. On the other hand the lack of rigidity results in a great diversity of degeneracy phenomena. To subject them to a systematic treatment it is convenient to start with the class HD of harmonic functions with finite Dirichlet integrals and the corresponding null class 0 HD The close connection with Dirichlet’s principle makes the class 0 HD the most significant one among degeneracy classes related to H.