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The Euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems. (English) Zbl 0782.30036
In this interesting paper the author organizes and extends his earlier work on the imbedding of open Riemann surfaces in compact Riemann surfaces [Hiroshima Math. J. 14, 371-399 (1984; Zbl 0567.30033); Trans. Am. Math. Soc. 301, 299-311 (1987; Zbl 0626.30046); Holomorphic functions and moduli I, Proc. Workshop, Berkeley/Calif. 1986, Publ. Math. Sci. Res. Inst. 10, 237-246 (1988; Zbl 0653.30028); Analytic functions theory of one complex variable, Pitman Res. Notes Math. Ser. 212, 287-298 (1989; Zbl 0682.30037)]. In particular he studies the moduli of (closed) tori into which a given open torus (open Riemann surface of genus one) can be imbedded subject to a normalization obtained by specifying canonical homology bases. The fundamental result is that the set of moduli forms a closed (possibly degenerate) disc in the upper half-plane. Its hyperbolic diameter is called the hyperbolic span of the given open torus. The author gives a variety of results in this connection and relations with results on maximal complementary area in the metric determined by a normaized holomorphic differential on the closed surface. He gives some analogues for plane domains including area results in the Euclidean and spherical metrics.

##### MSC:
 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30C35 General theory of conformal mappings 30F30 Differentials on Riemann surfaces
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##### References:
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