Homotopy of area decreasing maps by mean curvature flow

TL;DR

This paper proves that under certain curvature conditions, the mean curvature flow can smoothly deform an area decreasing map between Riemannian manifolds into a constant map, establishing a homotopy.

Contribution

It demonstrates that mean curvature flow induces a smooth homotopy to a constant map for area decreasing maps under natural curvature assumptions.

Findings

Mean curvature flow deforms area decreasing maps to constants

Homotopy exists under weak curvature conditions

Flow preserves smoothness during deformation

Abstract

Let $f:M\to N$ be a smooth area decreasing map between two Riemannian manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the curvatures of $(M,\gm)$ and $(N,\gn)$, we prove that the mean curvature flow provides a smooth homotopy of $f$ to a constant map.