Complete minimal surfaces and the puncture number problem

Abstract:

Given a nonnegative integer $g$, let $\mathcal {P}(g)$ denote the set of integers $N$ such that an arbitrary compact Riemann surface with genus $g$ can be completely conformally and minimally immersed in ${\mathbb {R}^3}$ (with finite total curvature) with exactly $N$ punctures. We prove that the infimum of $\mathcal {P}(g)$ is at most $4g$ and that the set $\mathcal {P}(g)$ may not miss any $3g$ consecutive integers larger than the infimum of $\mathcal {P}(g)$.