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/H is connected. The paper investigates the Ricci flow on a manifold M diffeomorphic to [0,1]×G/H. First, we prove a short-time existence and uniqueness theorem for a G-invariant solution g(t) satisfying the boundary condition II(g(t))=F(t,g∂M(t)) and the initial condition g(0)=gˆ. Here, II(g(t)) is the second fundamental form of ∂M, g∂M is the metric induced on ∂M by g(t), F is a smooth map and gˆ is a metric on M. Second, we study Perelman's F-functional on M. Our results show, roughly speaking, that F is non-decreasing on a G-invariant solution to the modified Ricci flow, provided that this solution satisfies boundary conditions inspired by a paper of Gianniotis.