View source for Gromov's compactness theorem (geometry)

{{Short description|On when a set of compact Riemannian manifolds of a given dimension is relatively compact}}
{{distinguish|text = [[Gromov's compactness theorem (topology)|Gromov's compactness theorem in symplectic geometry]]}}
In the [[mathematical]] field of [[metric geometry]], [[Mikhael Gromov (mathematician)|Mikhael Gromov]] proved a fundamental '''compactness theorem''' for sequences of [[metric space]]s. In the special case of [[Riemannian manifolds]], the key assumption of his compactness theorem is automatically satisfied under an assumption on [[Ricci curvature]]. These theorems have been widely used in the fields of [[geometric group theory]] and [[Riemannian geometry]].

==Metric compactness theorem==
The [[Gromov–Hausdorff metric|Gromov–Hausdorff distance]] defines a notion of distance between any two [[metric space]]s, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as [[Gromov–Hausdorff convergence]]. Gromov found a condition on a sequence of [[compactness (topology)|compact]] metric spaces which ensures that a [[subsequence]] converges to some metric space relative to the Gromov–Hausdorff distance:{{sfnm|1a1=Bridson|1a2=Haefliger|1y=1999|1loc=Theorem 5.41|2a1=Burago|2a2=Burago|2a3=Ivanov|2y=2001|2loc=Theorem 7.4.15|3a1=Gromov|3y=1981|3loc=Section 6|4a1=Gromov|4y=1999|4loc=Proposition 5.2|5a1=Petersen|5y=2016|5loc=Proposition 11.1.10}}
<blockquote>Let {{math|(''X''<sub>''i''</sub>, ''d''<sub>''i''</sub>)}} be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number {{math|ε}} there is a [[natural number]] {{mvar|N}} and, for every {{mvar|i}}, the set {{math|''X''<sub>''i''</sub>}} can be covered by {{mvar|N}} [[metric ball]]s of radius {{math|ε}}. Then the sequence {{math|(''X''<sub>''i''</sub>, ''d''<sub>''i''</sub>)}} has a subsequence which converges relative to the Gromov–Hausdorff distance.</blockquote>
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]].{{sfnm|1a1=Villani|1y=2009|1p=754}} Gromov first formally introduced it in his 1981 resolution of the [[Gromov's theorem on groups of polynomial growth|Milnor–Wolf conjecture]] in the field of [[geometric group theory]], where he applied it to define the [[Ultralimit|asymptotic cone]] of certain metric spaces.{{sfnm|1a1=Gromov|1y=1981|1loc=Section 7|2a1=Gromov|2y=1999|2loc=Paragraph 5.7}} These techniques were later extended by Gromov and others, using the theory of [[ultrafilter]]s.{{sfnm|1a1=Bridson|1a2=Haefliger|1y=1999|1loc=Definition 5.50|2a1=Gromov|2y=1993|2loc=Section 2}}

==Riemannian compactness theorem==
Specializing to the setting of [[geodesically complete]] [[Riemannian manifold]]s with a fixed lower bound on the [[Ricci curvature]], the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the [[Bishop–Gromov inequality|Bishop–Gromov volume comparison theorem]]. As such, it follows that:{{sfnm|1a1=Gromov|1y=1999|1loc=Theorem 5.3|2a1=Petersen|2y=2016|2loc=Corollary 11.1.13}}
<blockquote>Consider a sequence of [[closed manifold|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.</blockquote>
The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure.{{sfnm|1a1=Gromov|1y=1999|1loc=Paragraph 5.5}} 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==
{{reflist}}
'''Sources.'''
{{refbegin}}
*{{cite book|mr=1744486|last1=Bridson|first1=Martin R.|last2=Haefliger|first2=André|title=Metric spaces of non-positive curvature|series=Grundlehren der mathematischen Wissenschaften|volume=319|publisher=[[Springer-Verlag]]|location=Berlin|year=1999|isbn=3-540-64324-9|doi=10.1007/978-3-662-12494-9|author-link1=Martin Bridson|author-link2=André Haefliger|zbl=0988.53001}}
*{{wikicite|ref={{sfnRef|Burago|Burago|Ivanov|2001}}|reference={{cite book|mr=1835418|zbl=0981.51016|last1=Burago|first1=Dmitri|last2=Burago|first2=Yuri|last3=Ivanov|first3=Sergei|title=A course in metric geometry|series=[[Graduate Studies in Mathematics]]|volume=33|publisher=[[American Mathematical Society]]|location=Providence, RI|year=2001|isbn=0-8218-2129-6|author-link1=Dmitri Burago|author-link2=Yuri Burago|author-link3=Sergei Ivanov (mathematician)|doi=10.1090/gsm/033|ref=none}} {{erratum|https://www.pdmi.ras.ru/~svivanov/papers/bbi-errata.pdf|checked=yes}}}}
*{{cite journal|author-link1=Mikhael Gromov (mathematician)|last1=Gromov|first1=Mikhael|title=Groups of polynomial growth and expanding maps|journal=[[Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'Institut des Hautes Études Scientifiques]]|volume=53|year=1981|pages=53–73|zbl=0474.20018|doi=10.1007/BF02698687|url=http://www.numdam.org/item/?id=PMIHES_1981__53__53_0|mr=0623534|s2cid=121512559 }}
*{{cite conference|last1=Gromov|first1=M.|author-link1=Mikhael Gromov (mathematician)|title=Asymptotic invariants of infinite groups|zbl=0841.20039|book-title=Geometric group theory. Vol. 2|conference=Symposium held at Sussex University (Sussex, July 1991)|pages=1–295|series=London Mathematical Society Lecture Note Series|issue=182|publisher=[[Cambridge University Press]]|location=Cambridge|year=1993|mr=1253544|isbn=0-521-44680-5|editor-last1=Niblo|editor-first1=Graham A.|editor-last2=Roller|editor-first2=Martin A.}}
*{{cite book|author-link1=Mikhael Gromov (mathematician)|last1=Gromov|first1=Misha|title-link=Metric Structures for Riemannian and Non-Riemannian Spaces|title=Metric structures for Riemannian and non-Riemannian spaces|edition=Based on the 1981 French original|series=Progress in Mathematics|volume=152|publisher=[[Birkhäuser|Birkhäuser Boston, Inc.]]|location=Boston, MA|year=1999|isbn=0-8176-3898-9|mr=1699320|translator-last1=Bates|translator-first1=Sean Michael|others=With appendices by [[Mikhail Katz|M. Katz]], [[Pierre Pansu|P. Pansu]], and [[Stephen Semmes|S. Semmes]].|doi=10.1007/978-0-8176-4583-0|zbl=0953.53002}}
*{{cite book|last1=Petersen|first1=Peter|title=Riemannian geometry|edition=Third edition of 1998 original|series=[[Graduate Texts in Mathematics]]|volume=171|publisher=[[Springer Publishing|Springer, Cham]]|year=2016|isbn=978-3-319-26652-7|mr=3469435|doi=10.1007/978-3-319-26654-1|zbl=1417.53001}}
*{{cite book|mr=2459454|last1=Villani|first1=Cédric|title=Optimal transport. Old and new|series=Grundlehren der mathematischen Wissenschaften|volume=338|publisher=[[Springer-Verlag]]|location=Berlin|year=2009|isbn=978-3-540-71049-3|author-link1=Cedric Villani|doi=10.1007/978-3-540-71050-9|zbl=1156.53003}}
{{refend}}

{{Manifolds}}
{{Metric spaces}}

[[Category:Differential geometry]]
[[Category:Theorems in Riemannian geometry]]