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

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.