The Boundary Conjecture for Leaf Spaces
Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 2941-2950.

We prove that the boundary of an orbit space or more generally a leaf space of a singular Riemannian foliation is an Alexandrov space in its intrinsic metric, and that its lower curvature bound is that of the leaf space. A rigidity theorem for positively curved leaf spaces with maximal boundary volume is also established and plays a key role in the proof of the boundary problem.

On montre que le bord d’un espace d’orbites, ou plus généralement l’espace quotient d’un feuilletage riemannien singulier, est un espace d’Alexandrov muni de sa distance intrinsèque, et que la borne inférieure de sa courbure coincide avec celle de l’espace des feuilles. On établit aussi un théorème de rigidité pour les espaces de feuilles de courbure strictement positive maximisant le volume de leur bord, qui joue un rôle clef dans la preuve du théorème du bord.

Published online:
DOI: 10.5802/aif.3341
Classification: 53C23, 53C12, 53C24, 51K10
Keywords: Alexandrov Geometry, Singular Riemannian Foliations, Leaf Spaces, Lens Charaterization
Mot clés : Geométrie d’Alexandrov, feuilletage riemannien singulier, l’espace des feuilles, caractérisation des lentilles
Grove, Karsten 1; Moreno, Adam 1; Petersen, Peter 2

1 Department of Mathematics University of Notre Dame Notre Dame, IN 46556 (USA)
2 Department of Mathematics UCLA Los Angeles, CA 90095 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Grove, Karsten and Moreno, Adam and Petersen, Peter},
     title = {The {Boundary} {Conjecture} for {Leaf} {Spaces}},
     journal = {Annales de l'Institut Fourier},
     pages = {2941--2950},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {7},
     year = {2019},
     doi = {10.5802/aif.3341},
     language = {en},
     url = {}
AU  - Grove, Karsten
AU  - Moreno, Adam
AU  - Petersen, Peter
TI  - The Boundary Conjecture for Leaf Spaces
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 2941
EP  - 2950
VL  - 69
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  -
DO  - 10.5802/aif.3341
LA  - en
ID  - AIF_2019__69_7_2941_0
ER  - 
%0 Journal Article
%A Grove, Karsten
%A Moreno, Adam
%A Petersen, Peter
%T The Boundary Conjecture for Leaf Spaces
%J Annales de l'Institut Fourier
%D 2019
%P 2941-2950
%V 69
%N 7
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.3341
%G en
%F AIF_2019__69_7_2941_0
Grove, Karsten; Moreno, Adam; Petersen, Peter. The Boundary Conjecture for Leaf Spaces. Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 2941-2950. doi : 10.5802/aif.3341.

[1] Alexander, Stephanie; Kapovitch, Vitali; Petrunin, Anton An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds, Ill. J. Math., Volume 52 (2008) no. 3, pp. 1031-1033 | DOI | MR | Zbl

[2] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001 | MR

[3] Burago, Yuri; Gromov, Mikhail; Perelʼman, Gregory A.D. Alexandrov spaces with curvature bounded below, Russ. Math. Surv., Volume 47 (1992) no. 2, pp. 1-58 | DOI

[4] Grove, Karsten; Petersen, Peter A Lens Rigidity Theorem in Alexandrov Geometry (2018) (

[5] Hang, Fengbo; Wang, Xiaodong Rigidity theorems for compact manifolds with boundary and positive Ricci curvature, J. Geom. Anal., Volume 19 (2009) no. 3, pp. 628-642 | DOI | MR | Zbl

[6] Lytchak, Alexander; Thorbergsson, Gudlaugur Curvature explosion in quotients and applications, J. Differ. Geom., Volume 85 (2010), pp. 117-139 | DOI | MR | Zbl

[7] Mendes, Ricardo; Radeschi, Marco A slice theorem for singular Riemannian foliations, with applications, Trans. Am. Math. Soc., Volume 371 (2019) no. 7, pp. 4931-4949 | DOI | MR | Zbl

[8] Molino, Pierre Riemannian foliations, Progress in Mathematics, 73, Birkhäuser, 1988 | MR | Zbl

[9] Münzner, Hand F. Isoparametrische Hyperflächen in Sphären, Math. Ann., Volume 251 (1980) no. 1, pp. 57-71 | DOI | Zbl

[10] Petersen, Peter Riemannian geometry, 171, Springer, 2016 | MR | Zbl

[11] Petrunin, Anton Applications of quasigeodesics and gradient curves, Comparison geometry (1997), pp. 203-219 | Zbl

[12] Petrunin, Anton A globalization for non-complete but geodesic spaces, Math. Ann., Volume 366 (2016) no. 1-2, pp. 387-393 | DOI | MR | Zbl

[13] Radeschi, Marco Lecture notes on singular Riemannian foliations (

Cited by Sources: