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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Meromorphic functions on a compact Riemann surface and associated complete minimal surfaces

Author: Kichoon Yang
Journal: Proc. Amer. Math. Soc. 105 (1989), 706-711
MSC: Primary 53A10; Secondary 30F10
MathSciNet review: 953749
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Abstract: We prove that given any meromorphic function $ f$ on a compact Riemann surface $ M'$ there exists another meromorphic function $ g$ on $ M'$ such that $ \left\{ {df,g} \right\}$ is the Weierstrass pair defining a complete conformal minimal immersion of finite total curvature into Euclidean $ 3$-space defined on $ M'$ punctured at a finite set of points. As corollaries we obtain i) any compact Riemann surface can be immersed in Euclidean $ 3$-space as in the above with at most $ 4p + 1$ punctures, where $ p$ is the genus of the Riemann surface; ii) any hyperelliptic Riemann surface of genus $ p$ can be so immersed with at most $ 3p + 4$ punctures.

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PII: S 0002-9939(1989)0953749-1
Article copyright: © Copyright 1989 American Mathematical Society