Study of boundary behavior of holomorphic functions and harmonic functions

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1.Riemann surfaces(1)A sufficient condition was given for certain infinitely sheeted covering surfaces of the complex plane in order to possess no Green's functions.(2)For hyperbolic (i.e.possessing Green's function) Riemann surfaces R, it was shown that there exists no unbounded positive harmonic functions on R if and only if the Martin boundary of R consists of finitely many minimal points and each minimal point has positive harmonic measure. Moreover, for hyperbolic Riemann surfaces R, it was shown that there exists no Dirichlet infinite positive harmonic functions on R if and only if the Martin boundary of R consists of finitely many minimal points, each minimal point has positive harmonic measure and each minimal function is Dirichlet finite.(3)It was shown that, if 2 or 3-shetted unlimited covering surfaces R and S of the complex plane are quasiconformal equivalent to each other, then the harmonic dimension of R is equal to that of S.2.Picard Principle(1)It was shown that the Picard dimension of a rotation free hyperbolic signed Radon measure dm is equal to that of a measure obtained by adding a fundamental measure to dm.(2)For Green's function with respect to a staionary invariant Schoredinger equation with a Radon measure as potential, generalizations of Ohtsuka's theorem, Herve's theorem and Lahtinen' theorem are given.3.Bounded holomorphic functions(1)It was shown that a sequence in a plane domain satisfying a certain geometric condition is interpolating if the closures of two parts of the sequence is mutually disjoint.(2)It was shown that if the fiber of the maximal ideal space over a regular boundary point of a plane domain is not a peak set, then there exists a sequence converging to the boundary point which is not interpolating but harmonic interpolating.