Helicoidal minimal surfaces of prescribed genus, I

For every genus g, we prove that S^2 x 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 Euclidean 3-space R^3 that are helicoidal at infinity. In a companion paper, we prove that helicoidal surfaces in R^3 of every prescribed genus occur as such limits of examples in S^2 x R.