Short-time existence of the Ricci flow on noncompact Riemannian manifolds

Xu, Guoyi

doi:10.1090/s0002-9947-2013-05998-3

Cited by 20 publications

(19 citation statements)

References 14 publications

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“…When (M, g 0 ) is a complete noncompact Riemannian manifold, similar results were obtained by Ye Li [13] and G. Xu [16]. The assumptions of [13] is much weaker than (1.1) and (1.2) in case n = 4.…”

Section: )supporting

confidence: 73%

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Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Yang

2011

Ann Glob Anal Geom

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Section: )supporting

confidence: 73%

“…studied how to deform the metric on closed manifolds [6]. Also one can deform a metric locally by using the local Ricci flow [11,12,13,16,18]. Throughout this paper, we use Rm(g) and Ric(g) to denote the Riemannian curvature tensor and Ricci tensor with respect to the metric g respectively.…”

Section: Introductionmentioning

confidence: 99%

Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Yang

2011

Ann Glob Anal Geom

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“…In the paper [21], the author extends the results of Yang to the case that the manfiold is non-compact, and Ricci ≥ −1 and an L p bound on the curvature holds (p > (n/2)) (see also [34]).…”

Section: Previous Resultsmentioning

confidence: 96%

Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below

Simon¹

2011

Journal für die reine und angewandte Mathematik (Crelles Journa

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We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M, g) such that: a) (M, g) is non-collapsed (i.e. the volume of an arbitrary ball of radius one is bounded from below by v > 0 ) b) the Ricci curvature of (M, g) is bounded from below by k, c) the geometry at infinity of (M, g) is not too extreme (or (M, g) is compact). Given such initial data (M, g) we show that a Ricci flow exists for a short time interval [0, T ), where T = T (v, k) > 0. This enables us to construct a Ricci flow of any (possibly singular) metric space (X, d) which arises as a Gromov-Hausdorff limit of a sequence of 3-manifolds which satisfy a), b) and c) uniformly. As a corollary we show that such an X must be a manifold. This shows that the conjecture of M.Anderson-J.Cheeger-T.Colding-G.Tian is correct in dimension three.* part of this work was completed during the author's stay at Universität Münster in the semester 2008/09. This work was partially supported by SFB/Transregio 71.

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“…Proof. By [23], the Ricci flow with initial data g 0 has short time solution g(t) so that the curvature has the following bound:…”

Section: )mentioning

confidence: 99%

“…Acknowledgement: The second author would like to thank Albert Chau for some usefully discussions and for bringing our attention to the results in [23].…”

Section: Introductionmentioning

confidence: 99%

Kähler-Ricci flow with unbounded curvature

Huang¹,

Tam²

2018

American Journal of Mathematics

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Let g(t) be a complete solution to the Ricci flow on a noncompact manifold such that g(0) is Kähler . We prove that if |Rm(g(t))| g(t) ≤ a/t for some a > 0, then g(t) is Kähler for t > 0. We prove that there is a constant a(n) > 0 depending only on n such that the following is true: Suppose g(t) is a complete solution to the Kähler-Ricci flow on a noncompact n-dimensional complex manifold such that g(0) has nonnegative holomorphic bisectional curvature and such that |Rm(g(t))| g(t) ≤ a(n)/t, then g(t) has nonnegative holomorphic bisectional curvature for t > 0. These generalize the results in [21]. As corollaries, we prove that (i) any complete noncompact Kähler manifold with nonnegative complex sectional curvature with maximum volume growth is biholomorphic to C n ; and (ii) there is ǫ(n) > 0 depending only on n such that if (M n , g 0 ) is a complete noncompact Kähler manifold of complex dimension n with nonnegative holomorphic bisectional curvature and maximum volume growth and if (1 + ǫ(n)) −1 h ≤ g 0 ≤ (1 + ǫ(n))h for some Riemannian metric h with bounded curvature, then M is biholomorphic to C n .

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Short-time existence of the Ricci flow on noncompact Riemannian manifolds

Cited by 20 publications

References 14 publications

Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below

Kähler-Ricci flow with unbounded curvature

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