Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow

TL;DR

This paper studies the evolution of hypersurfaces under mean curvature flow within a background manifold evolving by Ricci flow, showing conditions for convergence to round points or geodesic spheres.

Contribution

It establishes new convergence results for hypersurfaces under combined mean curvature and Ricci flow conditions with initial pinching assumptions.

Findings

Hypersurfaces shrink to round points under certain conditions.

Hypersurfaces converge to totally geodesic spheres with suitable initial pinching.

Results depend on initial pinching conditions of the manifold and hypersurface.

Abstract

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the background manifold is sufficiently pinched and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.