Minimal immersions of punctured compact Riemann surfaces in ${\bf R}\sp 3$

Abstract

We prove that a hyperelliptic Riemann surface of genus $g$ can be completely conformally and minimally immersed in ${R^3}$ with finite total curvature with at most $3g + 2$ punctures; an arbitrary compact Riemann surface of genus $g$ can be so immersed with at most $4g$ punctures. Moreover, we show that there is at least a one-parameter family of nonisometric such immersions for a given punctured compact Riemann surface. Our results improve earlier results of Gackstatter-Kunert and the author.