Helicoidal minimal surfaces of prescribed genus

Abstract

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of examples in $S^2\times R$.

87 pages. This paper combines and supersedes our earlier two papers, Helicoidal minimal surfaces of prescribed genus, I and II [arXiv:1304.5861 and arXiv:1304.6180]. This version (16 November, 2016) fixes a few typos