Uniqueness of Solutions of Ricci Flow on Complete Noncompact Manifolds

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures. Recently there is a lot of study on the Ricci flow on manifolds by R. Hamilton [H1–6], S.Y. Hsu [Hs1–7], B. Kleiner and J. Lott [KL], J. Morgan and G. Tian [MT],G . Perelman [P1], [P2], P. Daskalopoulos, L. Ji, N. Sesum [DS], [JS], W.X. Shi [S1], [S2], R. Ye [Ye] and others. We refer the readers to the lecture notes by B. Chow [C] and the book [CK] by B. Chow and D. Knopf on the basics of Ricci flow and the papers [P1], [P2] of G. Perelman for the most recent results on Ricci flow. Existence of solution (M, g(t)), 0 ≤ t ≤ T , of the Ricci flow equation ∂ ∂t gij = −2Rij (0.1) on compact manifold M where Rij(t) is the Ricci curvature of g(t) and gij(x, 0) = gij(x) is a smooth metric on M is proved by R. Hamilton in [H1]. R. Hamilton [H1] also proved that when gij(x) is a metric of strictly positive Ricci curvature, then the evolving metric will converge modulo scaling to a metric of constant positive curvature. Similiar result was obtained by R. Hamilton [H2] for compact 4-dimensional manifolds with positive curvature operator. Harnack inequality for the Ricci flow was proved by R. Hamilton 1991 Mathematics Subject Classification. Primary 58J35, 53C43 Secondary 35K55.