Graphical mean curvature flow with bounded bi-Ricci curvature

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 $BRi{c}_M$ of $M$ is bounded from below by the sectional curvature $\ddot{I}{ƒ}_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ri{c}_M\ge \sup \{0,{\sup }_N\ddot{I}{ƒ}_N\}$. These results significantly improve known results on the graphical mean curvature flow in codimension $2$.