Math 2.0

Differential geometry Metric and synthetic methods in global geometry Theorem

Gromov's compactness theorem

Builds on Gromov–Hausdorff convergenceRiemannian manifoldRicci curvatureBishop–Gromov inequalityTotally bounded setAlexandrov space

Statement

Let n≥2n \ge 2 be an integer, let kk be a real number and let D>0D > 0. Let ℳ\mathcal{M} be the class of closed Riemannian manifolds MM of dimension nn whose Ricci curvature satisfies Ric⁡≥(n−1)k\operatorname{Ric} \ge (n - 1) k and whose diameter is at most DD. Then every sequence in ℳ\mathcal{M} has a subsequence that converges in the Gromov–Hausdorff sense to a compact metric space.

Remarks

The proof rests on a criterion of Gromov. A class of compact metric spaces is precompact for dGHd_{GH} when the diameters are bounded and, for each ε>0\varepsilon > 0, the number of ε\varepsilon-balls needed to cover each space is bounded. This is a uniform version of being totally bounded. For the manifolds in ℳ\mathcal{M}, the Bishop–Gromov inequality bounds the number of disjoint balls of radius ε/2\varepsilon/2, and hence the number of covering balls of radius ε\varepsilon.

The limit space need not be a manifold, and its dimension may drop. For example, the flat tori ℝ2/(ℤ×1mℤ)\R^2 / (\Z \times \tfrac{1}{m}\Z) have Ric⁡=0\operatorname{Ric} = 0 and bounded diameter, and they converge to a circle of length 11 as m→∞m \to \infty. Without the lower curvature bound the statement fails. Take spheres with mm long thin spikes of length 11 whose tips are pairwise at distance at least 22. The diameters stay bounded, but at least mm balls of radius 12\tfrac{1}{2} are needed to cover, so no subsequence converges.

Going deeper

For a student

This is the starting point of the study of limit spaces of manifolds with Ricci curvature bounded below. Under a lower bound on sectional curvature the limits are Alexandrov spaces. The theorem is used to prove finiteness and pinching results by contradiction: take a bad sequence, pass to a limit, and study the limit.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Alexandrov space, concept, no evidence levelAlexandrov spaceconceptGromov–Hausdorff convergence, concept, no evidence levelGromov–Hausdorff conv…conceptRicci curvature, concept, no evidence levelRicci curvatureconceptRiemannian distance, concept, no evidence levelRiemannian distanceconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSpace form, concept, no evidence levelSpace formconceptTotally bounded set, concept, no evidence levelTotally bounded setconceptBishop–Gromov inequality, statement, no evidence levelBishop–Gromov inequal…statementGromov's compactness theorem, statement, no evidence levelGromov's compactness …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Alexandrov space concept
  2. Gromov–Hausdorff convergence concept
  3. Ricci curvature concept
  4. Riemannian distance concept
  5. Riemannian manifold concept
  6. Space form concept
  7. Totally bounded set concept
  8. Bishop–Gromov inequality statement
  9. Gromov's compactness theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Gromov%27s_compactness_theorem_(geometry)
About this page
TagYGSKZFTEX1
URIm2://YGSKZFTEX1
Kindstatement, theorem
WritingW0 stub
MSC53C23, 53C21
StatusThis copy of the site has no build data. Run m2 status YGSKZFTEX1.
AboutGromov–Hausdorff convergence
UsesGromov–Hausdorff convergence, Riemannian manifold, Ricci curvature, Bishop–Gromov inequality, Totally bounded set, Alexandrov space
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Changed2 November 2026

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