Gromov's compactness theorem (geometria)

Gromov's compactness theorem (geometria) Not to be confused with Gromov's compactness theorem in symplectic geometry.

In the mathematical field of metric geometry, Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. 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 distance defines a notion of distance between any two metric spaces, 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 compact metric spaces which ensures that a subsequence converges to some metric space relative to the Gromov–Hausdorff distance:[1] Deixar (XI, di) be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number ε there is a natural number N and, for every i, the set Xi can be covered by N metric balls of 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 geodesically complete Riemannian manifolds 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 volume comparison theorem. As such, segue que:[5] 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, Teorema 5.41; Burago, Burago & Ivanov 2001, Teorema 7.4.15; Gromov 1981, Seção 6; Gromov 1999, Proposição 5.2; Petersen 2016, Proposição 11.1.10. ^ Villani 2009, p. 754. ^ Gromov 1981, Seção 7; Gromov 1999, Paragraph 5.7. ^ Bridson & Haefliger 1999, Definição 5.50; Gromov 1993, Seção 2. ^ Gromov 1999, Teorema 5.3; Petersen 2016, Corolário 11.1.13. ^ Gromov 1999, Paragraph 5.5.

Fontes.

Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Noções básicas de ciências matemáticas. Volume. 319. Berlim: Springer-Verlag. doi:10.1007/978-3-662-12494-9. ISBN 3-540-64324-9. SENHOR 1744486. Zbl 0988.53001. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001). A course in metric geometry. Pós Graduação em Matemática. Volume. 33. Providência, RI: Sociedade Americana de Matemática. doi:10.1090/gsm/033. ISBN 0-8218-2129-6. SENHOR 1835418. Zbl 0981.51016. (Erratum: [1]) Gromov, Mikhael (1981). "Groups of polynomial growth and expanding maps". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 53: 53-73. doi:10.1007/BF02698687. SENHOR 0623534. Zbl 0474.20018. Gromov, M. (1993). "Asymptotic invariants of infinite groups". In Niblo, Graham A.; Rolo, Martin A. (ed.). Teoria dos grupos geométricos. Volume. 2. Symposium held at Sussex University (Sussex, Julho 1991). Série de notas de palestras da London Mathematical Society. Cambridge: Cambridge University Press. pp. 1-295. ISBN 0-521-44680-5. SENHOR 1253544. Zbl 0841.20039. Gromov, Misha (1999). Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Volume. 152. Translated by Bates, Sean Michael. With appendices by M. Katz, P. Pansu, and S. Semmes. (Based on the 1981 French original ed.). Boston, MA: Birkhäuser Boston, Inc. doi:10.1007/978-0-8176-4583-0. ISBN 0-8176-3898-9. SENHOR 1699320. Zbl 0953.53002. Petersen, Peter (2016). Geometria Riemanniana. Textos de Graduação em Matemática. Volume. 171 (Terceira edição do 1998 original ed.). Springer, Cham. doi:10.1007/978-3-319-26654-1. ISBN 978-3-319-26652-7. SENHOR 3469435. Zbl 1417.53001. Vilões, Cédric (2009). Optimal transport. Old and new. Noções básicas de ciências matemáticas. Volume. 338. Berlim: Springer-Verlag. doi:10.1007/978-3-540-71050-9. ISBN 978-3-540-71049-3. SENHOR 2459454. Zbl 1156.53003. ocultar manifolds vte (Glossário) Conceitos básicos Atlas de variedades topológicas Variedades diferenciadas/suaves Estrutura diferencial Atlas suaveSubvariedade Variedades riemannianasMapa suaveSubmersãoPushforwardEspaço tangenteForma diferencialCampo vetorial Principais resultados (Lista) Índice de Atiyah–SingerDarboux'sDe Rham'sFrobeniusGeneralized StokesHopf–RinowNoether'sSard'sWhitney incorporação Mapas CurvaDifeomorfismo LocalGeodésicoMapa exponencial na teoria de LieFoliaçãoImersãoCurva integral Derivada de LieSeçãoSubmersão Tipos de variedades Fechadas(Quase) Complexo(Quase) ContatoFiberedFinslerFlatG-structureHadamardHermitianHyperbolicKählerKenmotsuLie group Lie álgebraVariedade com limiteOrientadoParalelizávelPoissonPrimeQuaternionicHypercomplex(Pseudo−, Sub-) Riemannian Rizza(Quase) SymplecticTame Tensors Vectors DistributionLie bracketPushforwardTangent space bundleTorsionVector fieldVector flow Covectors Closed/ExactCovariant derivadoCotangent space bundleDe Rham cohomologyForma diferencialValor vetorialDerivado exteriorProduto interiorPullbackRicci curvature flowRiemann curvature tensorRiemann densidade do campoTensor formaVolume product Wedge Bundles AdjointAffineAssociatedCotangentDualFiber(Companhia) álgebra FibrationJetLie(Estábulo) NormalPrincipalSpinorSubbundleTangentTensorVector Connections AffineCartanEhresmannFormGeneralizedKoszulLevi-CivitaPrincipalVectorTransporte paralelo Relacionado Classificação de variedadesGauge theoryHistoryMorse theoryMoving frameMorse theory Generalizations Banach variedadeDiffeologyDiffietyFréchet variedadeK-theoryOrbifoldCálculo secundário sobre álgebras comutativasShe afEstratifoldSupermanifoldCategorias de espaço estratificado topologicamente: Differential geometryTheorems in Riemannian geometry