Gromov's compactness theorem (geometry)
In the
Metric compactness theorem
The
Let (Xi, di) be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number ε there is a
metric ballsof radius ε. Then the sequence (Xi, di) has a subsequence which converges relative to the Gromov–Hausdorff distance.
The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the Arzelà–Ascoli theorem in the theory of uniform convergence.[2] Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of geometric group theory, where he applied it to define the asymptotic cone of certain metric spaces.[3] These techniques were later extended by Gromov and others, using the theory of ultrafilters.[4]
Riemannian compactness theorem
Specializing to the setting of
Consider a sequence of closed Riemannian manifolds with a uniform lower bound on the Ricci curvature and a uniform upper bound on the diameter. Then there is a subsequence which converges relative to the Gromov–Hausdorff distance.
The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure.[6] This special case of the metric compactness theorem is significant in the field of Riemannian geometry, as it isolates the purely metric consequences of lower Ricci curvature bounds.
References
- ^ Bridson & Haefliger 1999, Theorem 5.41; Burago, Burago & Ivanov 2001, Theorem 7.4.15; Gromov 1981, Section 6; Gromov 1999, Proposition 5.2; Petersen 2016, Proposition 11.1.10.
- ^ Villani 2009, p. 754.
- ^ Gromov 1981, Section 7; Gromov 1999, Paragraph 5.7.
- ^ Bridson & Haefliger 1999, Definition 5.50; Gromov 1993, Section 2.
- ^ Gromov 1999, Theorem 5.3; Petersen 2016, Corollary 11.1.13.
- ^ Gromov 1999, Paragraph 5.5.
Sources.
- Zbl 0988.53001.
- Zbl 0981.51016. (Erratum: [1])
- Zbl 0474.20018.
- Zbl 0841.20039.
- Zbl 0953.53002.
- Petersen, Peter (2016). Riemannian geometry. Zbl 1417.53001.
- Zbl 1156.53003.