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Meromorphic Functions on a Compact Riemann Surface and Associated Complete Minimal Surfaces

1989
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Proceedings of the American Mathematical Society
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We prove that given any meromorphic function / on a compact Riemann surface M' there exists another meromorphic function g on M' such that {df, g} is the Weierstrass pair defining a complete conformai 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

doi:10.2307/2046924
fatcat:65mogoyrnbc5di3fxzxb7qzyfy