A Lower Bound for the Scalar Curvature of the Standard Solution of the Ricci Flow

In this paper we will give a rigorous proof of the lower bound for the scalar curvature of the standard solution of the Ricci flow conjectured by G. Perelman. We will prove that the scalar curvature R of the standard solution satisfies R(x, t) ≥ C0/(1−t) ∀x ∈ R , 0 ≤ t < 1, for some constant C0 > 0. Recently there is a lot of study of Ricci flow on manifolds by R. Hamilton [H1-6], S.Y. Hsu [Hs1-6], P. Lu and G. Tian [LT], G. Perelman [P1], [P2], W.X. Shi [S1], [S2], L.F. Wu [W1], [W2], and others. In [H1] R. Hamilton studied the Ricci flow on compact manifolds with strictly positive Ricci curvature. He proved that if the metric g(x, t) of the manifold evolves by the Ricci flow, ∂ ∂t gij = −2Rij (0.1) with gij(x, 0) = gij(x), then the evolving metric will converge modulo scaling to a metric of constant positive curvature. This result was later extended to compact four dimensional manifold with positive curvature operator by R. Hamilton [H2] and to non-compact complete manifolds by W.X. Shi [S1], [S2]. Behaviour and properties of Ricci flow on R are also studied by P. Daskalopoulos and M.A. Del Pino [DP], S.Y. Hsu [Hs1-4] and L.F. Wu [W1], [W2]. We refer the reader to the survey paper of R. Hamilton [H5] for previous results on Ricci flow, the lecture notes [Ch] of B. Chow and the book [CK] of B. Chow and D. Knopf for the most recent results on Ricci flow. 1991 Mathematics Subject Classification. Primary 58J35, 53C44 Secondary 58C99.