Hamilton’s compactness theorem for Ricci flow

E889125

compactness theorem mathematical theorem result in geometric analysis

Hamilton’s compactness theorem for Ricci flow is a fundamental result in geometric analysis that provides conditions under which a sequence of Ricci flows on Riemannian manifolds subconverges to a limiting Ricci flow, enabling powerful compactness and convergence arguments in the study of geometric evolution.

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Observed surface forms (1)

Surface form	Occurrences
The formation of singularities in the Ricci flow	1

Statements (46)

Predicate	Object
instanceOf	compactness theorem ⓘ mathematical theorem ⓘ result in geometric analysis ⓘ
appearsIn	Hamilton’s papers on three-manifolds with positive Ricci curvature ⓘ Hamilton’s work on four-manifolds with positive curvature operator ⓘ
appliesTo	Ricci flow NERFINISHED ⓘ Riemannian manifolds ⓘ
assumes	completeness of the Riemannian metrics ⓘ lower bounds on injectivity radius at base points ⓘ solutions defined on a common time interval ⓘ uniform curvature bounds on space-time regions ⓘ
category	theorem about geometric evolution equations ⓘ
concludes	existence of a subsequence converging to a limiting Ricci flow ⓘ limit is a complete solution of the Ricci flow ⓘ pointed convergence of manifolds with base points ⓘ smooth Cheeger–Gromov convergence on compact subsets ⓘ
dimension	valid in arbitrary dimension ⓘ
field	Riemannian geometry NERFINISHED ⓘ differential geometry ⓘ geometric analysis ⓘ
framework	pointed Riemannian manifolds with base points ⓘ time-dependent Riemannian metrics ⓘ
generalizes	compactness results for static Riemannian manifolds ⓘ
historicalContext	developed in the 1980s and 1990s ⓘ
implies	existence of geometric limits for controlled sequences of flows ⓘ precompactness of families of Ricci flows under given bounds ⓘ
importance	enables passage to limits in sequences of evolving geometries ⓘ fundamental tool in modern Ricci flow theory ⓘ
influenced	later compactness theorems for other geometric flows ⓘ
namedAfter	Richard S. Hamilton NERFINISHED ⓘ
provides	compactness for families of Ricci flows ⓘ conditions for subsequential convergence of Ricci flows ⓘ
relatedTo	Cheeger–Gromov compactness theorem NERFINISHED ⓘ Hamilton’s Ricci flow with surgery NERFINISHED ⓘ Perelman’s work on Ricci flow and the Poincaré conjecture ⓘ
requires	uniform bounds on all covariant derivatives of curvature on compact time intervals ⓘ
technicalTool	Arzelà–Ascoli type arguments for tensor fields ⓘ Shi’s derivative estimates for Ricci flow ⓘ harmonic coordinate estimates ⓘ
usedFor	convergence arguments in Ricci flow ⓘ extracting convergent subsequences of Ricci flows ⓘ geometric evolution equations ⓘ
usedIn	analysis of singularity formation in Ricci flow ⓘ blow-up analysis near singularities ⓘ classification of singularity models ⓘ construction of ancient solutions via limits ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Richard S. Hamilton → knownFor → Hamilton’s compactness theorem for Ricci flow ⓘ

Richard S. Hamilton → notableWork → Hamilton’s compactness theorem for Ricci flow ⓘ

this entity surface form: The formation of singularities in the Ricci flow