Generalized convexity and surfaces of negative curvature

Convexity estimates for surfaces moving by curvature functions

We consider the evolution of compact surfaces by fully nonlinear, parabolic curvature ows for which the normal speed is given by a smooth, degree one homogeneous function of the principal curvatures of the evolving surface. Under no further restrictions on the speed function, we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow. This ge...

Curvature and Convexity I

Complete surfaces with negative extrinsic curvature

N. V. Efimov [Efi64] proved that there is no complete, smooth surface in R with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if M has sectional curvature between two constants K2 and K3, then there exists K1 < min(K2, 0) such that M contains no smooth, complete immersed surface with curvature below K1. Optimal values of K1 are dete...

Monge-ampère Equations and Surfaces with Negative Gaussian Curvature

In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as ...

Shadows and Convexity of Surfaces

We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satis es a certain two-piece-property, and solve the \shadow problem" formulated by H. Wente.