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

Kichoon Yang
1989 Proceedings of the American Mathematical Society  
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
more » ... ce; ii) any hyperelliptic Riemann surface of genus p can be so immersed with at most 3p + 4 punctures.
doi:10.2307/2046924 fatcat:65mogoyrnbc5di3fxzxb7qzyfy