Anisotropic flow of convex hypersurfaces by the square root of the scalar curvature

Abstract

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor ψ when the strictly convex initial hypersurface in Euclidean space is suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. For a general case in dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.