Convergence and rigidity of manifolds under Ricci curvature bounds

Anderson, Michael T.

doi:10.1007/bf01233434

Cited by 306 publications

(483 citation statements)

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“…Hence K g = o(r −2 ) and diameter is controlled from above and volume growth is controlled from below (from the Sobolev inequality) on each annulus M 2kr − M k −1 r . We can then infer from Anderson-Cheeger harmonic radius' theory [2,3,16,23] that the rescaled annuli (M kri − M k −1 ri , r −2 i g) are covered by a finite (and uniformly bounded) number of balls of uniformly bounded size where the metric coefficients are C 1,α -close to the euclidean metric. Hence one gets volume growth control from above.…”

Section: Vol 77 (2002)mentioning

confidence: 99%

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The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Comment. Math. Helv.

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Abstract.It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite L n/2 -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Mathematics Subject Classification (2000). 53C21, 58J60.

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Section: Vol 77 (2002)mentioning

confidence: 99%

“…Our manifolds will satisfy finiteness of the L n/2 -norm of Ricci curvature (1.1), together with an adequate Sobolev inequality: 2) which is the extra hypothesis we choose for this case. For technical reasons, we also have to add some mild assumption on scalar curvature:…”

Section: R(x) ρ} ρ Dρ < ∞mentioning

confidence: 99%

The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Comment. Math. Helv.

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“…On the other hand, by Lemma 2.10, since we have a curvature bound sup M Rm ≤ AF 1 6 ≤ 1, if we choose ι sufficiently small with respect to δ we obtain…”

Section: Proofs Of Smoothing Theoremsmentioning

confidence: 99%

“…(1) r δ ≥ ρ, (2) F (g) ≤ ǫ, the L 2 flow with initial condition g exists on [0, ρ 4 ] and moreover satisfies the estimates (1) |Rm| gt ≤ AF 1 6 (g t )t − 1 2 , (2) inj gt ≥ ιt 1 4 , (3) diam gt ≤ 2(ρ + diam g 0 ).…”

Section: Statement Of Singularity Decompositionmentioning

confidence: 99%

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A Concentration‐Collapse Decomposition for L² Flow Singularities

Streets¹

2014

Comm Pure Appl Math

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Abstract. We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the L 2 curvature flow and Calabi flow, in dimensions n ≤ 4. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small energy and a volume growth lower bound, in the vein of Perelman's pseudolocality result. Next, we generalize our technique from prior work to exhibit local smoothing estimates for the L 2 flow in the presence of a curvature-related bound. A final key ingredient is a new local ǫ-regularity result for L 2 -critical metrics with possibly nonconstant scalar curvature. Applications of these results include new compactness and diffeomorphism-finiteness theorems for smooth compact four-manifolds satisfying the necessary and effectively minimal hypotheses of L 2 curvature pinching and a volume noncollapsing condition.

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Space of Ricci Flows I

Chen

Wang

2012

Comm Pure Appl Math

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In this paper, we study the moduli spaces of m‐dimensional, κ‐noncollapsed Ricci flow solutions with bounded $\int |Rm|^{{m}/{2}}$ and bounded scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study the estimates of isoperimetric constants, the Kähler‐Ricci flows, and the moduli spaces of gradient shrinking solitons. © 2012 Wiley Periodicals, Inc.

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Convergence and rigidity of manifolds under Ricci curvature bounds

Cited by 306 publications

References 20 publications

The Huber theorem for non-compact conformally flat manifolds

The Huber theorem for non-compact conformally flat manifolds

A Concentration‐Collapse Decomposition for L² Flow Singularities

Space of Ricci Flows I

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Convergence and rigidity of manifolds under Ricci curvature bounds

Cited by 306 publications

References 20 publications

The Huber theorem for non-compact conformally flat manifolds

The Huber theorem for non-compact conformally flat manifolds

A Concentration‐Collapse Decomposition for L2 Flow Singularities

Space of Ricci Flows I

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Product

Resources

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A Concentration‐Collapse Decomposition for L² Flow Singularities