Contraction of convex hypersurfaces in Riemannian spaces

This paper concerns the deformation of hypersurfaces in Riemannian spaces using fully nonlinear parabolic equations defined in terms of the Weingarten curvature. It is shown that any initial hypersurface satisfying a natural convexity condition produces a solution which converges to a single point in finite time, and becomes spherical as the limit is approached. The result has topological implications including a new proof of the 1/4-pinching sphere theorem of Klingenberg, Berger, and Rauch, and a new "dented sphere theorem" which allows some negative curvature.This paper concerns the deformation of hypersurfaces in Riemannian spaces using fully nonlinear parabolic equations defined in terms of the Weingarten curvature. It is shown that any initial hypersurface satisfying a natural convexity condition produces a solution which converges to a single point in finite time, and becomes spherical as the limit is approached. The result has topological implications including a new proof of the 1/4-pinching sphere theorem of Klingenberg, Berger, and Rauch, and a new "dented sphere theorem" which allows some negative curvature.