arXiv math/97122351997-12math.GT

The compression theorem I

Colin Rourke, Brian Sanderson

备注：: This is a shortened version of "The compression theorem": applications have been omitted and will be published as part III. For a preliminary version of part III, see section 5 onwards of version 2 of this paper. This version (v3) is published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper14.abs.html
发表信息：: Geom. Topol. 5 (2001) 399-429
DOI：: 10.2140/gt.2001.5.399

英文摘要：

This the first of a set of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer-Verlag (1986); 2.4.5 C'] and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding. In the second paper in the series we give a proof in the spirit of Gromov's proof and in the third part we give applications.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The compression theorem II: directed embeddings

This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.

2001-04-24math.GT

The Gromov--Guth--Whitney embedding theorem

This is an appendix to arXiv:1610.04888, "Quantitative null-cobordism", which improves one of the main results of that paper to a near-sharp one. It is not a self-contained paper.

2018-05-16math.GT

关于Gromov非嵌入定理

On the Gromov non-embedding theorem

本文提出新方法，给出Gromov关于Hölder嵌入到Heisenberg群不存在性定理的推广的初等证明。

2023-12-21math.DGmath.GT

The Gromov-Eliashberg theorem by microlocal sheaf theory

The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally microsupports of sheaves. We explain how we can deduce the Gromov-Eliashberg theorem from the involutivity theorem of Kashiwara and Schapira which says that the microsupport of a sheaf is coisotropic.

2013-11-01math.GTmath.SG

Entropy, Weil-Petersson translation distance and Gromov norm for surface automorphisms

Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. We also prove that the Weil-Petersson translation distance does the same from both sides in general. The proofs are in fact immediately derived from the theorem of Brock together with some other strong theorems and small observations.

2011-10-07math.GT

Energy of harmonic functions and Gromov's proof of Stallings' theorem

We provide the details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions.

2007-07-28math.DGmath.GRmath.GT

更多相关论文收起更多论文

分层空间的Gromov-Tischler定理

The Gromov-Tischler theorem for stratified spaces

本文将Gromov-Tischler定理从流形推广到分层空间：定义分层空间上的辛结构，并证明具有整上同调类的辛分层空间可辛嵌入到配备标准Kähler形式的复射影空间中。

2023-08-13math.GTmath.SG

From Seiberg-Witten to Gromov: MCE and Singular Symplectic Forms

Motivated by various possible generalizations of Taubes's $SW=Gr$ theorem [T] to Floer-theoretic setting, we prove certain variants of Taubes's convergence theorem in \cite{T} (the first part of his proof of $SW=Gr$). In place of the closed symplectic 4-manifold considered in [T], this article considers non-compact manifolds with cylindrical ends, equipped with a self-dual harmonic 2-form with non-degenerate zeroes. This extends and simplifies some central technical ingredients of the author's prior work in [LT] and [KLT5]. Other expected applications include: extending the $HM=PFH$ theo…

2023-01-10math.GT

Holonomic approximation and Gromov's h-principle

In 1969 M. Gromov in his PhD thesis greatly generalized Smale-Hirsch-Phillips immersion-submersion theory by proving what is now called the h-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generality. Gromov's thesis was brought to the West by A. Phillips and was popularized in his talks. However, most western mathematicians first learned about Gromov's theory from A. Haefliger's article. The current paper is devoted to the same subject as the papers of Gromov and Haeflige…

2001-01-24math.GTmath.SG

Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theorem, part of Epstein's local structure theorem for foliations by compact leaves, and a continuity theorem of Alvarez and Candel. Several examples are discussed.

2013-04-23math.DGmath.GT

精细曲线图的Gromov边界的连通性

Connectedness of the Gromov boundary of fine curve graphs

本文研究了高亏格闭曲面精细曲线图的Gromov边界拓扑，证明了有界测地线像定理及其非紧性，并得出该边界关于某些视觉度量是线性连通的。

2025-03-03math.GTmath.MG

Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free $δ$-hyperbolic marked groups whose entropy is bounded above by a number $H$ is finite with cardinality depending on $δ$ and $H$ alone. From these results, we draw homo…

2021-11-08math.GRmath.GT

Gromov compactness theorem for stable curves

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', in particular they can be non-closed and the complex structures on them can vary arbitrarily. In connection with this, we study in §2 moduli spaces of nodal curves with boundary and define a natural complex structure for such moduli spaces. We obtain an apriori estimate for pseud…

1999-03-08math.CVmath.DGmath.GT

The compression theorem III: applications

This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts (math.GT/9712235 and math.GT/0003026) gave proofs. Here we are concerned with applications. We give short new (and constructive) proofs for immersion …

2003-09-25math.ATmath.GT

小尺度指标理论、数量曲率与 Gromov 的单纯范数

Small scale index theory, scalar curvature, and Gromov's simplicial norms

本文研究具有数量曲率下界流形的拓扑复杂度，提出小尺度指标定理，利用泛覆盖的数量曲率、体积和单射半径界建立 Gromov 单纯范数上界，推广了 Lichnerowicz 消没定理并类比 Cheeger 有限性定理。

2025-11-03math.DGmath.GTmath.KT

具积分加权里奇曲率界的芬斯勒流形上的博内-迈尔斯定理

The Bonnet-Myers theorem on Finsler manifolds with integral weighted Ricci curvature bounds

该研究在芬斯勒度量测度流形上，推导了受积分加权里奇曲率控制的体积比较定理，建立非同心球的Bishop-Gromov体积比较定理，进而证明了具积分加权里奇曲率界的芬斯勒流形上的博内-迈尔斯型定理。

2026-09-03math.DGmath.GT

Gromov-Hausdorff 距离、Borsuk-Ulam 定理与 Vietoris-Rips 复形

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

本文利用 Vietoris-Rips 复形与等变拓扑方法，将不连续函数畸变与 Gromov-Hausdorff 距离联系起来，推广了球面间奇映射的 Borsuk-Ulam 型结果，并改进不同维数球面间 Gromov-Hausdorff 距离的上下界。

2025-07-05math.ATmath.GTmath.MG