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On a result of M. Heins

Published online by Cambridge University Press:  20 January 2009

James A. Jenkins
James A. Jenkins
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey 08540 Washington UniversitySt Louis, Missouri 63130
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Some years ago Heins (1) proved that a Riemann surface which can be conformally imbedded in every closed Riemann surface of a fixed positive genus g is conformally equivalent to a bounded plane domain. In the proof the main effort is required to prove that a surface satisfying this condition is schlichtartig. Heins gave quite a simple proof of the remaining portion (1; Lemma 1). The main part of the proof depended on exhibiting a family of surfaces of genus g such that a surface which could be conformally imbedded in all of them was necessarily schlichtartig. Another proof using a different construction was recently given by Rochberg (2). We will give here a further proof based on the method of the extremal metric and using a further construction which is in some ways more direct than those previously given.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 4 , September 1975 , pp. 371 - 373
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Heins, M., A problem concerning the continuation of Riemann surfaces, Contributions to the Theory of Riemann Surfaces (Annals of Mathematics Studies, No. 30, Princeton University Press, 1953), 55–62.Google Scholar
(2) Rochberg, R., Continuation of Riemann surfaces, to appear.Google Scholar