On the Gauss Curvature of Minimal Surfaces!?)

1. Summary of results. The following is known: let 5 be a minimal surface defined by z=f(x, y) over the region D:x2+y20 there exists a surface with W=l such that \K\ =64/9a*2. It is further shown that among surfaces with W=l the slightly stronger inequality \K\ S6i/9d2 holds, and by the above example this is best possible. All of these results are extended to cases where 5 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 ol p two complex variables f and w, such that the correspondence between if and w is analytic, and for the Gauss curvature K at each point of N we have the formula