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

TL;DR

This paper proves the existence and convergence of an anisotropic curvature flow for convex hypersurfaces, showing they evolve smoothly to round spheres or solitons depending on the dimension and rescaling.

Contribution

It establishes the existence, smoothness, and convergence of a new anisotropic curvature flow driven by the square root of scalar curvature for convex hypersurfaces.

Findings

Flow exists smoothly for convex hypersurfaces under certain conditions.

Rescaled surfaces converge to round spheres in higher dimensions.

In dimension two, the limit profile satisfies a soliton equation.

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 $\psi$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. In dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.