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Perelman's reduced volume and a gap theorem for the Ricci flow

2009
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Communications in analysis and geometry
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In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricciflat manifolds. As a corollary, a gap theorem for gradient shrinking Ricci solitons is also obtained. We say that (M, g(τ )) is ancient when g(τ ) exists for all τ ∈ [0, ∞). Ancient solutions are important objects in the study

doi:10.4310/cag.2009.v17.n2.a3
fatcat:zqj563eazzek7mz654fsl7hl6a