Negatively curved sets on surfaces of constant mean curvature in R3 are large

It is proved that the negatively curved set M_ on a nonparametric surface M of constant mean curvature in R-3 must extend to the boundary partial derivative M, if M_ is nonempty. For M parametric, if M_ is compactly included in the interior of M; then M_ is at least as large as an extremal domain. The results imply certain convexity results on elliptic partial differential equations. Second-order calculus of variation is employed.