A CONVEX SURFACE WITH FRACTAL CURVATURE

Mean Curvature Flow with Convex Gauss Image

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence results in both cases. We also study the asymptotic behavior of these soluti...

Convex tours of bounded curvature

We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with n vertices. We present an O(m + n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest po...

Convex Hulls of Bounded Curvature

In this paper, we consider the problem of computing a convex hull of bounded curvature of a set S of points in the plane, i.e. a set containing S and whose boundary is a curve of bounded curvature of minimal length. We prove that, if the radius of the smallest disk that contains S is greater than 1, such a hull is unique. We show that the computation of a convex hull of bounded curvature reduce...

The Existence of Convex Body with Prescribed Curvature Measures

Metrics with Non-negative Ricci Curvature on Convex Three-manifolds

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path-connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the threeball) is contractible. As an application, using results of Maximo, Nunes, and Smith [MNS], we show the existence of properly embedd...