On the Gauss curvature of minimal surfaces

Abstract

1. Summary of results. The following is known: let S be a minimal surface defined by z-=f(x, y) over the region D: x2+y2 O there exists a surface with W= 1 such that K Kj 64/9d2. It is further shown that among surfaces with W=1 the slightly stronger inequality |K| <64/9d2 holds, and by the above example this is best possible. All of these results are extended to cases where S is not representable in the form z=f(x, y). The methods used also yield a number of other results for the class of surfaces considered. 2. Introduction. The majority of results in this paper are derived from the following observation. Given a point p on a minimal surface, one may assign to each point of a suitable neighborhood N of p two complex variables r and w, such that the correspondence between r and w is analytic, and for the Gauss curvature K at each point of N we have the formula