The Ricci Flow on Domains in Cohomogeneity One Manifolds

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G\mathrm{/}H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G\mathrm{/}H$. First, we prove a short-time existence and uniqueness theorem for a $G$-invariant solution $g(t)$ satisfying the boundary condition $(g(t))=F(t,g_{\mathrm{\partial }M}(t))$ and the initial condition $g(0)=\widehat{g}$. Here, $(g(t))$ is the second fundamental form of $\mathrm{\partial }M$, $g_{\mathrm{\partial }M}$ is the metric induced on $\mathrm{\partial }M$ by $g(t)$, $F$ is a smooth map and $\widehat{g}$ is a metric on $M$. Second, we study Perelman’s $\mathcal{F}$-functional on $M$. Our results show, roughly speaking, that $\mathcal{F}$ is non-decreasing on a $G$-invariant solution to the modified Ricci flow, provided that this solution satisfies boundary conditions inspired by the 2012 paper of Gianniotis.