Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

TL;DR

This paper studies a coupled geometric and diffusion system involving scaled mean curvature flow, analyzing qualitative properties like surface area decrease, convexity preservation, and self-intersection development, with insights into its gradient flow structure.

Contribution

It introduces a coupled mean curvature flow and diffusion system, analyzing its qualitative properties and providing examples of long-term behavior and geometric evolution.

Findings

Surface area decreases strictly over time.

Mean convexity is preserved, convexity is not.

Existence of a hypersurface developing self-intersection.

Abstract

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.