Forced Convex Mean Curvature Flow in Euclidean Spaces

TL;DR

This paper studies the behavior of convex hypersurfaces evolving under mean curvature flow with a forcing term, revealing conditions for shrinking, expanding, or converging to spheres, and extending classical flow results.

Contribution

It introduces a generalized mean curvature flow with forcing in Euclidean spaces, analyzing long-term behavior and convergence, extending prior classical and volume-preserving flows.

Findings

Flow shrinks to a point with small forcing term.

Flow exists indefinitely and expands with large forcing term.

Flow converges to a sphere under specific forcing and initial conditions.

Abstract

In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Huisken's well-known mean curvature flow and McCoy's mixed volume preserving mean curvature flow.