Cheeger–Gromov compactness theorem

E888038

compactness theorem mathematical theorem

The Cheeger–Gromov compactness theorem is a fundamental result in Riemannian geometry that gives conditions under which a sequence of Riemannian manifolds has a subsequence converging (in the Gromov–Hausdorff or smooth sense) to a limit space.

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Observed surface forms (1)

Surface form	Occurrences
Gromov’s precompactness theorem	1

Statements (45)

Predicate	Object
instanceOf	compactness theorem ⓘ mathematical theorem ⓘ
appliesTo	compact Riemannian manifolds ⓘ families of manifolds with bounded geometry ⓘ pointed complete Riemannian manifolds ⓘ
assumes	uniform bounds on curvature and its derivatives in C^k or C^∞ versions ⓘ uniform lower bound on Ricci curvature or sectional curvature ⓘ uniform lower bound on injectivity radius in smooth versions ⓘ uniform upper bound on diameter in many formulations ⓘ
concerns	Gromov–Hausdorff convergence ⓘ precompactness in appropriate topology ⓘ sequences of Riemannian manifolds ⓘ smooth convergence of Riemannian manifolds ⓘ
field	Riemannian geometry ⓘ differential geometry ⓘ metric geometry ⓘ
givesConditionFor	Gromov–Hausdorff precompactness ⓘ existence of convergent subsequence of Riemannian manifolds ⓘ smooth Cheeger–Gromov convergence ⓘ
guarantees	subsequence convergence under uniform geometric bounds ⓘ
implies	existence of convergent subsequence of manifolds under curvature and diameter bounds ⓘ limit is a Riemannian manifold in smooth convergence setting ⓘ limit is a compact metric space in Gromov–Hausdorff setting ⓘ
limitObject	Riemannian manifold in smooth Cheeger–Gromov sense ⓘ compact metric space in Gromov–Hausdorff sense ⓘ
namedAfter	Jeff Cheeger NERFINISHED ⓘ Mikhael Gromov NERFINISHED ⓘ
relatedTo	Arzelà–Ascoli theorem NERFINISHED ⓘ Gromov compactness theorem NERFINISHED ⓘ precompactness of isometry classes of manifolds ⓘ
typicalHypothesis	fixed dimension of the manifolds ⓘ uniform bound on absolute value of sectional curvature ⓘ uniform positive lower bound on injectivity radius ⓘ
usedIn	Ricci flow theory NERFINISHED ⓘ geometric analysis ⓘ global Riemannian geometry ⓘ study of collapsing Riemannian manifolds ⓘ study of moduli spaces of Riemannian metrics ⓘ
usesConcept	C^k convergence of Riemannian metrics ⓘ Gromov–Hausdorff distance NERFINISHED ⓘ Ricci curvature NERFINISHED ⓘ curvature tensor ⓘ injectivity radius ⓘ pointed Gromov–Hausdorff convergence ⓘ sectional curvature ⓘ

Referenced by (2)

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Hamilton’s compactness theorem → relatedTo → Cheeger–Gromov compactness theorem ⓘ

this entity surface form: Gromov’s precompactness theorem