Graphical mean curvature flow with bounded bi-Ricci curvature

TL;DR

This paper proves long-time existence and convergence of the graphical mean curvature flow for area decreasing maps between certain Riemannian manifolds, under bounded bi-Ricci curvature conditions, improving previous results in codimension 2.

Contribution

It establishes new long-time existence and convergence results for the graphical mean curvature flow with bounded bi-Ricci curvature, extending prior work in codimension 2.

Findings

Flow preserves strictly area decreasing property under curvature bounds

Flow converges smoothly to a minimal map when Ricci curvature condition is met

Results improve known theorems in codimension 2

Abstract

We consider the graphical mean curvature flow of strictly area decreasing maps $f:M\to N$, where $M$ is a compact Riemannian manifold of dimension $m>1$ and $N$ a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature $BRic_M$ of $M$ is bounded from below by the sectional curvature $\sigma_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_N\sigma_N\}$. These results significantly improve known results on the graphical mean curvature flow in codimension $2$.