论文标题
图形平均曲率流具有有界双层曲率的曲率
Graphical mean curvature flow with bounded bi-Ricci curvature
论文作者
论文摘要
我们考虑严格区域的图形平均曲率流量减少地图$ f：m \至n $，其中$ m $是尺寸的紧凑riemannian歧管$ m> 1 $ and $ n $ a $ n $是一个完整的里曼尼亚人的界面表面。我们证明了该流量的长期存在，并且保留了严格的区域降低属性，当Bi-ricci曲率$ bric_m $ $ m $的$ m $从下面界定为$ n $的截面曲率$σ_n$。此外，如果$ ric_m \ ge \ sup \ {0，{\ sup}_nσ_n\} $，我们将获得平滑收敛到最小地图。这些结果显着改善了Codimension $ 2 $中图形平均曲率流的已知结果。
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 $σ_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_Nσ_N\}$. These results significantly improve known results on the graphical mean curvature flow in codimension $2$.
