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Cited by 2 publications
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We study conformal embeddings of a noncompact Riemann surface of finite genus into compact Riemann surfaces of the same genus and show some of the close relationships between the classical theory of univalent functions and our results. Some new problems are also discussed. This article partially intends to introduce our results and to invite the function-theorists on plane domains to the topics on Riemann surfaces. KEYWORDS: univalent functions, Riemann surfaces, continuations of a Riemann surface, span IntroductionThe rôle of univalent functions in complex analysis is doubtlessly important. Various problems concerning the coefficients of the seties expansion have been above all deeply studied and many remarkable results are now in our possession. Univalent function theory is still today an active field. The domain of definition of a univalent function is, by the definition, limited to a plane domain -a nonempty connected open subset of the Riemann sphere.Although it is desirable to study analyticity on general Riemann surfaces, univalent functions make sense only on planar Riemann surfaces. The main object of the present paper shall be hence univalent analytic mapping, i.e. conformal embedding of a Riemann surface into another. This causes however a new difficulty; we cannot designate the image of the given surface under such a mapping at the outset. In other words, we have to consider the following problem at first. What kind of Riemann surfaces do we have to choose as the image? The notion of continuations of a Riemann surface serves the purpose, as the discussion below shows.The first half of the present paper is devoted to a short summary of our basic results. In order to compare them with the classical theory of univalent functions, we confine ourselves to those topics whose counterpart can be easily found. (For other topics, see for instance, [38], [23], [39], and [40].) As an example of such topics we focus our attention to the span in the second half of the paper.The span is first defined by M. Schiffer ([30]) for multiply connected plain domains, and has been since then generalized in many different directions (see e.g., [15], [27], [28], [29]). Hamano-Maitani-Yamaguchi [10] considered a certain holomorphic family of finite Riemann surfaces parametrized by the unit disk D and showed that one of the generalized spans (the harmonic span in [28]) is a subharmonic function on D. Concerning the Schiffer span for plane domains a similar result is obtained by S. Hamano ([9]). The span with which we are concerned in this paper is, on the other hand, of different kind and presently defined only for Riemann surfaces of genus one, but shares many interesting function-theoretic properties with the Schiffer span (cf. [35], [36], [37], and [38]).We finally state our recent results which shows that our span is a natural object in the theory of functions of several complex variables. The details will appear elsewhere as a joint work with S. Hamano and H. Yamaguchi. Continuations of a Riemann SurfaceWe start with an ope...
We study conformal embeddings of a noncompact Riemann surface of finite genus into compact Riemann surfaces of the same genus and show some of the close relationships between the classical theory of univalent functions and our results. Some new problems are also discussed. This article partially intends to introduce our results and to invite the function-theorists on plane domains to the topics on Riemann surfaces. KEYWORDS: univalent functions, Riemann surfaces, continuations of a Riemann surface, span IntroductionThe rôle of univalent functions in complex analysis is doubtlessly important. Various problems concerning the coefficients of the seties expansion have been above all deeply studied and many remarkable results are now in our possession. Univalent function theory is still today an active field. The domain of definition of a univalent function is, by the definition, limited to a plane domain -a nonempty connected open subset of the Riemann sphere.Although it is desirable to study analyticity on general Riemann surfaces, univalent functions make sense only on planar Riemann surfaces. The main object of the present paper shall be hence univalent analytic mapping, i.e. conformal embedding of a Riemann surface into another. This causes however a new difficulty; we cannot designate the image of the given surface under such a mapping at the outset. In other words, we have to consider the following problem at first. What kind of Riemann surfaces do we have to choose as the image? The notion of continuations of a Riemann surface serves the purpose, as the discussion below shows.The first half of the present paper is devoted to a short summary of our basic results. In order to compare them with the classical theory of univalent functions, we confine ourselves to those topics whose counterpart can be easily found. (For other topics, see for instance, [38], [23], [39], and [40].) As an example of such topics we focus our attention to the span in the second half of the paper.The span is first defined by M. Schiffer ([30]) for multiply connected plain domains, and has been since then generalized in many different directions (see e.g., [15], [27], [28], [29]). Hamano-Maitani-Yamaguchi [10] considered a certain holomorphic family of finite Riemann surfaces parametrized by the unit disk D and showed that one of the generalized spans (the harmonic span in [28]) is a subharmonic function on D. Concerning the Schiffer span for plane domains a similar result is obtained by S. Hamano ([9]). The span with which we are concerned in this paper is, on the other hand, of different kind and presently defined only for Riemann surfaces of genus one, but shares many interesting function-theoretic properties with the Schiffer span (cf. [35], [36], [37], and [38]).We finally state our recent results which shows that our span is a natural object in the theory of functions of several complex variables. The details will appear elsewhere as a joint work with S. Hamano and H. Yamaguchi. Continuations of a Riemann SurfaceWe start with an ope...
Any open Riemann surface R 0 of finite genus g can be conformally embedded into a closed Riemann surface of the same genus, that is, R 0 is realized as a subdomain of a closed Riemann surface of genus g. We are concerned with the set M(R 0 ) of such closed Riemann surfaces. We formulate the problem in the Teichmüller space setting to investigate geometric properties of M(R 0 ). We show, among other things, that M(R 0 ) is a closed Lipschitz domain homeomorphic to a closed ball provided that R 0 is nonanalytically finite. MSC2020: 30Fxx, 32G15 Theorem 4. No Ioffe rays of M(R 0 ) hit M(R 0 ) again after departure. For every S ∈ T g \ M(R 0 ) there exists exactly one Ioffe ray of M(R 0 ) passing through S.Thus the complement T g \ M(R 0 ) is swept out by Ioffe rays of R 0 , plays a crucial role in proving Theorem 1. Moreover, Theorem 4 allows us to call M(R 0 ) close-to-convex.
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