Cheeger and Gromov's compactness theorem

The compactness theorem of Cheeger and Gromov , often also referred to as Gromov's compactness theorem or Gromov's precompactness theorem, is a mathematical theorem from the field of differential geometry . It makes a statement about the Gromov-Hausdorff convergence of sequences of Riemannian manifolds with given diameter, volume and curvature bounds. A direct consequence of the compactness theorem is Cheeger's finiteness theorem .

Compactness theorem

If it's for a series constants Riemannian manifolds , , are such that, for all the estimates ${\displaystyle (M_{i})_{i\in \mathbb {N} }}$ ${\displaystyle D}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle i\in \mathbb {N} }$

${\displaystyle \operatorname {diam} (M_{i})\leq D,\quad \operatorname {vol} (M_{i})\geq V,\quad -K\leq \sec(M_{i})\leq K}$

hold, then a subsequence in the Gromov-Hausdorff topology converges to a Riemann manifold . The volume , the diameter and the section curvatures denote the Riemann manifold . ${\displaystyle M_{i_{k}}}$ ${\displaystyle (M,g)}$ ${\displaystyle \operatorname {vol} (M_{i})}$ ${\displaystyle \operatorname {diam} (M_{i})}$ ${\displaystyle \sec(M_{i})}$ ${\displaystyle M_{i}}$

One can choose the partial sequence of Riemannian manifolds so that there are diffeomorphisms for which the Riemannian metric converges. ${\displaystyle (M_{i_{k}},g_{i_{k}})}$ ${\displaystyle \phi _{i_{k}}\colon M_{i_{k}}\to M}$ ${\displaystyle \phi _{i_{k}}^{*}g_{i_{k}}}$ ${\displaystyle g}$

literature

• Michail Leonidowitsch Gromow : Metric structures for Riemannian and non-Riemannian spaces. Based on the French original edition from 1981. With appendices by M. Katz, P. Pansu and S. Semmes. Translation from French by Sean Michael Bates. Progress in Mathematics, 152. Birkhauser Boston, Inc., Boston, MA, 1999. ISBN 0-8176-3898-9
• RE Greene, H. Wu: Lipschitz convergence of Riemannian manifolds. Pacific J. Math. 131 (1988) no. 1, 119-141.

Web links

• Peter Petersen : Convergence Theorems in Riemannian Geometry
• Robert Haslhofer: Convergence of Riemannian Manifolds