Uniform Sobolev Inequalities for Manifolds Evolving by Ricci Flow

Let M be a compact n-dimensional manifold, n ≥ 2, with metric g(t) evolving by the Ricci flow ∂gij/∂t = −2Rij in (0, T ) for some T ∈ R + ∪ {∞} with g(0) = g0. Let λ0(g0) be the first eigenvalue of the operator −∆g0 + R(g0) 4 with respect to g0. We extend a recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any n ≥ 2 when either T < ∞ or λ0(g0) > 0. As a consequence we extend Perelman’s local κ-noncollapsing result along the Ricci flow for any n ≥ 2 in terms of upper bound for the scalar curvature when either T < ∞ or λ0(g0) > 0. Recently there is a lot of studies on Ricci flow on manifolds because it is an important tool in understanding the geometry of manifolds [H1–3], [Hs1–3], [KL], [MT], [P1], [P2]. On the other hand given any compact n-dimensional manifold M , n ≥ 2, with a fixed metric g it is known that Sobolev inequalities hold [He]. More specifically for any q ∈ [1, n) and p satisfying 1 p = 1 q − 1 n (1) there exists a minimal constant Cp,q(M, g) > 0 such that the following holds (∫ M |u|dVg ) 1 p ≤ Cp,q(M, g) ((∫ M |∇u| dVg ) 1 q + 1 volg(M) 1 n (∫ M |u|dVg ) 1 q ) (2) 1991 Mathematics Subject Classification. Primary 58J35, 53C21 Secondary 46E35.