On semi-parabolic Riemann surfaces
Robert D. M. Accola · Transactions of the American Mathematical Society · 1963
A bordered Riemann surface whose double is parabolic will be called semi-parabolic.Denote the class of semi-parabolic surfaces by SO?.As usual, a compact bordered surface will be called a finite surface.In a sense, the interiors of semi-parabolic surfaces are the simplest hyperbolic surfaces since their hyperbolicity results entirely from the border which is given in their definition.On finite surfaces, the class of harmonic functions which are constant on each contour is a finite-dimensional vector space of functions with finite Dirichlet norm.This paper considers the corresponding class of functions on bordered surfaces of class SOg and generalizes some of the properties of harmonic measures on finite surfaces.In particular, for generalized harmonic measures, we investigate the level curves and their orthogonal trajectories.The principal results, Theorems 4.1 and 4.4, state, in a sense made precise, that almost all of the level curves of a generalized harmonic measure are analytic Jordan curves and almost all of their orthogonal trajectories begin and end on the border given in the definition of the surface.These results have application to the level curves of a Green's function via a theorem of Kuramochi.We also consider the question on a parabolic surface as to when a harmonic differential with finite norm and integral periods is a weak limit of period reproducing differentials. Definitions and notation.If FF is a Riemann surface, let F(W) stand for the Hubert space of square integrable differentials on W (2).For co,aer(W), let \\a\\w denote the norm of a, and (a, co)w the inner product(3).Let Tc and Te denote the closed and exact forms in T. Let Teo be the closure in Te of differentials of functions which vanish outside of compact sets.Define Tc0 = T*x' r* = Tc r\T* rh0 = r" nTC0 and rhe = rh nTe(4).rk is the Hubert space of Presented to the Society, October 22,1960, under the title Semi-parabolic Riemann surfaces; received by the editors May 3, 1962.(!) The work on this paper has been conducted over the last three years.During this time the author has been supported by the Office of Ordnance Research, U. S. Army, Contract number DA-19-020-ORD-3779 and by the Office of Naval Research, Contract number Nonr 562 (31).(2) For a complete discussion of the theory of square integrable differentials see Ahlfors and Sario[l, Chapter VJ. (3) In the notation T(W), \\a\\w, and (a, io)w the symbol W will be omitted if it is obvious from the context.(4) Tp is the set of differentials whose conjugates lie in rp.437