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 $${\mathbf{S}^2\times\mathbf{R}}$$ .