ON A CONJECTURE OF NITSCHE

We show that under the hypothesis of bounded Gaussian curvature, certain minimal surfaces are in fact of finite total curvature. We can then answer the following version of a conjecture of Nitsche (J. Math. Mech. 11 (1962), 295) under the hypothesis of bounded Gaussian curvature: Conjecture. Let M2 subset-of R3 be a complete minimal surface such that for some height function H, the level sets are (compact) Jordan curves. Then M is a catenoid.