A Riemann on the ideal boundary of a Riemann surface

1960

1. Various strides have been done to characterize the conformal structure of Riemann surfaces by the algebraic structure of some appropriate function algebras on them (cf. Bers [2], Rudin [29], Royden [26], [28], Heins [7], Kakutani [12], Wermer [33] etc.). In this paper we discuss, corresponding to the above, the problem to determine the quasiconformal structure of Riemann surfaces by the algebraic structure of some function algebras on them.

“…Some applications of Royden's algebra to the function theory were given by Royden [24], [25], [26], S. Mori [15], [16], Mori-Ota [17] and the present author [18], [19].…”

Section: )mentioning

confidence: 99%

Algebraic Criterion on Quasiconformal Equivalence of Riemann Surfaces

1960

“…Such u is determined uniquely by the distribution g('/) on I\{R), for functions in HBD takes its minimum and maximum on MR) (cf. S. Mori and M. Ota [7]). Thus HBD* is isometrically C-module isomorphic to the continuous function space CilΛR)) preserving the positiveness.…”

Section: A Riemann Surface R Belongs To the Class Mhbd* If And Only Imentioning

confidence: 95%

“…Lemma 1.6) u n (7.) of g n (7) converges uniformly to a function u in HBD*. Clearly u(Z) =g(Z) (Ze IΛR)).…”

Section: A Riemann Surface R Belongs To the Class Mhbd* If And Only Imentioning

confidence: 98%

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On a Ring Isomorphism Induced by Quasiconformal Mappings

1959

The purpose of this paper is to study the relationship between a certain isomorphism of some rings of functions on Riemann surfaces and a quasi-conformal mapping.It is well known that two compact Hausdorff spaces are topologically equivalent if and only if their rings of continuous functions are isomorphic. We shall establish an analougous result concerning a function ring on a Riemann surface and the quasi-conformal equivalence.

“…Let D be the double of D with respect to Γ. Then [5]). Any real-valued HBD-function defined on R takes its maximum and minimum on Δ.…”

Section: Harmonic Decomposition In This Section We Exclude the Trivmentioning

confidence: 99%

A Measure on the Harmonic Boundary of a Riemann Surface

1960

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.