Evolution of hypersurfaces under the scalar curvature flow

Abstract

We study the evolution of hypersurfaces whose rate of deformation in the inward normal direction is proportional to the scalar curvature at each point. We will present the $C^{\infty }$-regularity and preservation of strict convexity for hypersurfaces evolving under the scalar curvature flow before collapse. We also have $C^{1,1}$-optimal regularity for convex hypersurfaces with a flat spot and local upper and lower curvature bounds in the strictly convex region.