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Non-negative Ricci Curvature on Closed Manifolds under Ricci Flow
In this short paper we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wilking have for dimensions twelve and above. Moreover, the manifolds constructed here are...
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Theorem 2 Given n and v0 > 0, there exists ǫ0 > 0 depending only on n and v0 which has the following property. For any r0 > 0 and ǫ ∈ (0, ǫ0] suppose (Mn, g(t)) is a complete smooth solution to the Ricci flow on [0, (ǫr0) 2] with bounded sectional curvature, and assume that at t = 0 for some x0 ∈ M we have curvature bound |Rm |(x, 0) ≤ r 0 for all x ∈ Bg(0)(x0, r0), and volume lower bound Volg(...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2019
ISSN: 0001-8708
DOI: 10.1016/j.aim.2018.11.006