Helicoidal minimal surfaces of prescribed genus

Abstract

For every genus g, we prove that ${\mathbf{S}}^2\times \mathbf{R}$ 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 ${\mathbf{S}}^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in ${\mathbf{R}}^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in ${\mathbf{R}}^3$ of every prescribed genus occur as such limits of examples in ${\mathbf{S}}^2\times \mathbf{R}$.