arXiv:1304.6204v1 [math.DG] 23 Apr 2013 [email protected] nfrl one geometry bounded Uniformly 4 oitos rmvsaeadla ucin 4 functions leaf of Regularity 3 functions leaf and space Gromov , 2 Introduction 1 Contents ebsaiiyadteGoo-asofflimits Gromov-Hausdorff the and stability Reeb ∗ † . mohcmatesterm...... 14 . . 16 ...... 14 ...... convergence . . Gromov-Hausdorff vs . . Smooth theorem geometry compactness bounded smooth 4.3 uniformly A with of 4.2 Spaces 4.1 . euaiyterm 10 ...... 11 ...... stability . Reeb and . continuity . to Applications . theorems 3.2 Regularity 3.1 . xml:teRe rniin...... 9 . 5 . . 7 8 . 6 ...... 6 ...... transition . . . Reeb . . . . the . . . . Example: record . . broken . . . the . . 2.6 . Example: component . . Reeb . . the . . 2.5 . Example: cylinder . Reeb . the . foliation 2.4 Example: record . vinyl . the 2.3 . Example: . . 2.2 Definitions 2.1 MT autdd inis dlR g´ 25 10 Monte 11400 Igu´a 4225, UdelaR, Ciencias, de Facultad CMAT, PA nvri´ iree ai ui,4paeJsiu 75 Jussieu, place 4 Curie, Marie et Universit´e Pierre LPMA, o oitosb opc evs n otniytermo theorem continuity str a local and leaves, Epstein’s compact of by part foliations Corolla for theorem, Gromov-Hausdorff. stability merely local convergen of that Reeb’s instead show smooth also We wh is leaf cover. limit limiting the leaf’s of this space than covering a is foliation compact adl eea xmlsaediscussed. are examples Several Candel. eso htteGoo-asofflmto euneo leave of sequence a of limit Gromov-Hausdorff the that show We flae ncmatfoliations compact in leaves of etme 5 2018 25, September al Lessa Pablo Abstract 1 ∗ † 0 ai,France. Paris, 005 ie,Uuuy Email: Uruguay. video, cuetheorem ucture c sn larger no is ich f lae and Alvarez et uha such to ce ´ isinclude ries na in s 14 10 2 5 Smooth precompactness 17 5.1 Normsandsequentialcompactness ...... 18 5.2 Normalcoordinates...... 19 5.3 Transitionmaps ...... 19

6 Curvature and injectivity radius 22 6.1 Semicontinuity of the injectivity radius ...... 23

7 Smoothconvergenceandtensornorms 25 7.1 Coordinatefreedefinitionofconvergence...... 25 7.2 Convergenceoftensorfields ...... 27

8 Bounded geometry of leaves 29

9 Covering spaces and holonomy 29 9.1 Riemanniancoverings ...... 30 9.2 Holonomycovering ...... 30 9.3 Trivialholonomy ...... 31

10Convergenceofleafwisefunctions 33 10.1Adapteddistances ...... 33 10.2 Convergenceofleafwiseimmersions...... 35

1 Introduction

The Reeb local stability theorem [Ree47, Theorem 2] states that if the funda- mental group of a compact leaf in a foliation is finite then all nearby leaves are finite covers of it. It’s apparent from the proof that, besides compactness, the key property of the leaf which yields stability isn’t finiteness of its , but finiteness of its holonomy group. This gives rise to the standard generalization given for example in [CLN85, pg. 70]. In the special case when a leaf is compact and has trivial holonomy one can strengthen the conclusion to yield that all nearby leaves are diffeomorphic to the given leaf (see [Thu74, Theorem 2] where conditions under which one can guarantee trivial holonomy are discussed). Another context in which a Reeb-type stability result appeared is in the study of compact foliations by compact leaves (by which we mean that all leaves are compact). This family of foliations is surprisingly rich due to the fact that if the codi- mension is 3 or more there may be leaves with arbitrarily large volume (see [EV78]). However, if one assumes that there is a uniform upper bound for the volume of all leaves then Epstein’s local structure theorem [Eps76, Theorem 4.3] yields that each leaf has a neighborhood consisting of leaves which are finite covers of it. But, what can be said ab