Isotropy invariant graphical mean curvature flows in warped products

Abstract

In this paper, we study the graphical mean curvature flow in a warped product $_r G/K \times I$, where $G/K$ is a symmetric space of compact type, $I$ is an open interval, and $r$ is a smooth positive function on $I$. If the initial hypersurface is $K$-equivariant, then the $K$-equivariance is preserved along the mean curvature flow. Here, we note that isotropy group $K$ acts naturally on both $G/K$ and $_r G/K \times I$. If the flow is graphical, then it follows from the $K$-equivariance of the flow that it can be described by using $K$-invariant functions on $G/K$. We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that $G/K$ is a rank one symmetric space of compact type and the warping function $r$ satisfies certain additional properties. The proof is carried out by estimating the gradient of the $K$-invariant functions satisfying the flow equation.