The Boundary Conjecture for Leaf Spaces
Annales de l'Institut Fourier to appear, , 10 p.

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.

Classification:  53C23,  53C12,  53C24,  51K10
Keywords: Alexandrov Geometry, Singular Riemannian Foliations, Leaf Spaces, Lens Charaterization
@unpublished{AIF_0__0_0_A19_0,
     author = {Grove, Karsten and Moreno, Adam and Petersen, Peter},
     title = {The Boundary Conjecture for Leaf Spaces},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Grove, Karsten; Moreno, Adam; Petersen, Peter. The Boundary Conjecture for Leaf Spaces. Annales de l'Institut Fourier, to appear, 10 p.

[1] Alexander, Stephanie; Kapovitch, Vitali; Petrunin, Anton An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds, Ill. J. Math., Tome 52 (2008) no. 3, pp. 1031-1033 | Zbl 1200.53040

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

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

[4] Grove, Karsten; Petersen, Peter A Lens Rigidity Theorem in Alexandrov Geometry (2018) (https://arxiv.org/abs/1805.10221)

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

[6] Lytchak, Alexander; Thorbergsson, Gudlaugur Curvature explosion in quotients and applications, J. Differ. Geom., Tome 85 (2010), pp. 117-139 | Zbl 1221.53067

[7] Mendes, Ricardo; Radeschi, Marco A slice theorem for singular Riemannian foliations, with applications, Trans. Am. Math. Soc., Tome 371 (2019) no. 7, pp. 4931-4949 | Zbl 07050767

[8] Molino, Pierre Riemannian foliations, Progress in Mathematics, Tome 73, Birkhäuser, 1988 | Zbl 0633.53001

[9] Münzner, Hand F. Isoparametrische Hyperflächen in Sphären, Math. Ann., Tome 251 (1980) no. 1, pp. 57-71 | Zbl 0417.53030

[10] Petersen, Peter Riemannian geometry Tome 171, Springer, 2016

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

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

[13] Radeschi, Marco Lecture notes on singular Riemannian foliations (https://www.marcoradeschi.com)