A Caveat on the Convergence of the Ricci Flow for Pinched Negatively Curved Manifolds

where r = ∫ Rdμ/ ∫ dμ is the average scalar curvature (R is the scalar curvature) and Ric is the Ricci curvature tensor of h. Hamilton then spectacularly illustrated the success of this method by proving, when n = 3, that if the initial Riemannian metric has strictly positive Ricci curvature it evolves through time to a positively curved Einstein metric h∞ on M . And, because n = 3, such a Riemannian metric automatically has constant sectional curvature; hence (M, h∞) is a spherical space-form; i.e. its universal cover is the round sphere. Following Hamilton’s approach G. Huisken [6], C. Margerin [7] and S. Nishikawa [9] proved that, for every n, sufficiently pinched to 1 n-manifolds (the pinching constant depending only on the dimension) can be deformed, through the Ricci flow, to a spherical-space form.