Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products

Abstract

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On the classification of warped product Einstein metrics

“…When there is a critical point at t = 0, g S must be the round sphere to obtain a smooth metric, if there is no critical point, then g S is the metric of a level set of w. The result follows easily from these equations. For more details see, [7,11,20,35]. …”

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

“…Since ν M (t 0 ) > 0, it is a basic result in Alexandrov space theory (see for example Theorem 7.6 of [20]) that a subsequence of (M, g k (s), x 0 ) converges in the GromovHausdorff sense to an n-dimensional metric cone (M ,g(0), x 0 ) with vertex x 0 . By (6.3.1), the standard volume comparison and Theorem 4.2.2, we know that the injectivity radius of (M, g k (0)) at x k is uniformly bounded from below by a positive number ρ 0 .…”

Kähler manifolds with Ricci curvature lower bound

“…In [2], Cheeger and Colding choose the cut-off function to be a function of g. To make that work, the upper bound of a is needed. To avoid this problem, we define the cut-off function ϕ to be a function of r, explicitly,…”