Gauss curvature estimates for surfaces whose mean curvature is constant or has special properties

Abstract

Let z = w(x, y) represent an embedded (not necessarily simplyconnected), compact nonparametric surface in R 3 with mean curvature H, nonpositive Gauss curvature K. Set (−K) max , |H| max to be the global maximum of −K and H and set K 0 , H 0 to be the maximum15 π. In Part II, this result is extended to a special class of doubly-connected surfaces of constant mean curvature, whose boundary consists of two components, on one of which K = K 0 and on the other of which K = (−K) max .