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Uniqueness of solutions of Ricci flow on complete noncompact manifolds

By Shu-yu Hsu


Abstract. 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 [H1–3] and others. 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. Since the proof of existence of solution of the Ricci flow in [H1] is very hard, later D.M. DeTurck [D] deviced another method to prove existence and uniqueness of solution of Ricci flow on compact manifolds. Let M be a n-dimensional manifold with (M, gij(t)), 0 ≤ t ≤ T, being a solution of the Ricci flow (0.1) and let (N, hαβ) be a fixed n-dimensional manifold. He introduced the associated Ricci harmonic flow F: (M, g(t)) → (N, h) given by ∂F ∂t = ∆ g(t),hF (0.2

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Year: 2013
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