Abstract. We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.
1. Introduction 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow 6. The κ-noncollapsing estimate Appendix A. The logarithmic Sobolev inequalities on the euclidean space Appendix B. The estimate of e −tH Appendix C. From the estimate for e −tH to the Sobolev inequalityand all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the scalar curvature R g 0 , the modified Sobolev constantC S (M, g 0 ) (see Section 2 for its definition) and the volume vol g 0 (M) which are those of the initial metric g 0 . Consequently, there holds for each t ∈ [0, T ) M
The main results in this paper are: ( 1 ) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) L2-pinched Riemannian manifolds can be deformed to space forms through Ricci flow.
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