The Ricci Flow on Domains in Cohomogeneity One Manifolds

TL;DR

This paper proves short-time existence and uniqueness of $G$-invariant Ricci flow solutions on manifolds with boundary, and analyzes the behavior of Perelman's $$-functional under these flows with specific boundary conditions.

Contribution

It establishes a short-time existence and uniqueness theorem for $G$-invariant Ricci flows with boundary conditions and studies the monotonicity of Perelman's $$-functional in this setting.

Findings

Proved short-time existence and uniqueness of $G$-invariant Ricci flow solutions with boundary conditions.

Showed that Perelman's $$-functional is non-decreasing under the modified Ricci flow with certain boundary conditions.

Extended analysis of Ricci flow behavior to manifolds with boundary and symmetry group actions.

Abstract

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time existence and uniqueness theorem for a $G$-invariant solution $g(t)$ satisfying the boundary condition $\mathop{\mathrm{II}}(g(t))=F(t,g_{\partial M}(t))$ and the initial condition $g(0)=\hat g$. Here, $\mathop{\mathrm{II}}(g(t))$ is the second fundamental form of $\partial M$, $g_{\partial M}$ is the metric induced on $\partial M$ by $g(t)$, $F$ is a smooth map and $\hat g$ 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.