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For every genus g, we prove that S^2 x R contains complete, properlyembedded, genus-g minimal surfaces whose two ends are asymptotic to helicoidsof any prescribed pitch. We also show that as the radius of the S^2 tends toinfinity, these examples converge smoothly to complete, properly embeddedminimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In acompanion paper, we prove that helicoidal surfaces in R^3 of every prescribedgenus occur as such limits of examples in S^2 x R.

Helicoidal minimal surfaces of prescribed genus, I