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Difference between revisions of "Finite Riemann surface"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR>
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====Comments====
 
====Comments====
The notion of a finite Riemann surface should not be mixed up with that of a Riemann surface of finite type: A Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040320/f0403201.png" /> is of finite type if it can be imbedded in a compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040320/f0403202.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040320/f0403203.png" /> consists of finitely many points. Cf. also [[Riemann surfaces, classification of|Riemann surfaces, classification of]] and (the references to) [[Double of a Riemann surface|Double of a Riemann surface]].
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The notion of a finite Riemann surface should not be mixed up with that of a Riemann surface of finite type: A Riemann surface $M$ is of finite type if it can be imbedded in a compact Riemann surface $\tilde M$ such that $\tilde M \setminus M$ consists of finitely many points. Cf. also [[Riemann surfaces, classification of|Riemann surfaces, classification of]] and (the references to) [[Double of a Riemann surface|Double of a Riemann surface]].
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Revision as of 10:49, 18 October 2014

A Riemann surface of finite genus (cf. Genus of a surface) having finitely many non-degenerate boundary components. A finite Riemann surface can be imbedded in a closed Riemann surface — its double (cf. Double of a Riemann surface).

References

[1] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)


Comments

The notion of a finite Riemann surface should not be mixed up with that of a Riemann surface of finite type: A Riemann surface $M$ is of finite type if it can be imbedded in a compact Riemann surface $\tilde M$ such that $\tilde M \setminus M$ consists of finitely many points. Cf. also Riemann surfaces, classification of and (the references to) Double of a Riemann surface.

How to Cite This Entry:
Finite Riemann surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Finite_Riemann_surface&oldid=33781
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article