Contraction of convex surfaces by the Gauss curvature

Abstract

We review Ben Andrews’ article “Gauss Curvature Flow: the Fate of the Rolling Stones,” which was published on Inventiones Mathematicae in 1999. The purpose of this paper is to fill in technical aspects that Ben Andrews omitted in his paper, so that it can give more clear comprehension of his paper to the readers who have just begun studying the Gauss curvature flow. This paper focuses on smooth convex surfaces contracting in the direction of the normals by their Gauss curvature. By using various estimate methods such as the curvature estimate, the isoperimetric estimate, and the Harnack estimate, we prove that convex surfaces contracting by their Gauss curvature converge to a point in finite time, becoming spherical as they deform. Furthermore, the case where initial surfaces are non-smooth is considered. It is shown that the contraction of convex surfaces under the Gauss curvature flow is independent of the initial surfaces’ regularity conditions, which draws the conclusion that the same properties must be satisfied in the case of non-smooth initial surfaces.