Prescription de courbure des feuilles des laminations : retour sur un théorème de Candel
[Prescribing the curvature of leaves of laminations: revisiting a theorem by Candel]
Annales de l'Institut Fourier, Volume 71 (2021) no. 6, pp. 2549-2593.

In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger–Gromov topology on the space of complete pointed riemannian manifolds.

Dans cet article, nous revenons sur un célèbre théorème de Candel que nous renforçons en prouvant qu’étant donnée une lamination compacte par surfaces hyperboliques, toute fonction négative lisse dans les feuilles et transversalement continue est la fonction courbure d’une unique métrique laminée dans la classe conforme correspondante. Nous interprétons ce fait comme la continuité de solutions de certaines EDP elliptiques dans une topologie, dite de Cheeger–Gromov, sur l’espace des variétés riemanniennes complètes pointées.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3476
Classification: 57R30, 53C21, 30F10, 30F45
Mot clés : Laminations par surfaces hyperboliques, Prescription de courbure
Keywords: lamination by hyperbolic surfaces, prescrired curvature

Alvarez, Sébastien 1; Smith, Graham 2

1 CMAT, Facultad de Ciencias, Universidad de la República (Uruguay)
2 Instituto de Matemática, UFRJ, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, Cidade Universitária - Ilha do Fundão, Caixa Postal 68530, 21941-909, Rio de Janeiro, RJ (Brazil)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2021__71_6_2549_0,
     author = {Alvarez, S\'ebastien and Smith, Graham},
     title = {Prescription de courbure des feuilles des laminations~: retour sur un th\'eor\`eme de {Candel}},
     journal = {Annales de l'Institut Fourier},
     pages = {2549--2593},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {6},
     year = {2021},
     doi = {10.5802/aif.3476},
     zbl = {07554454},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3476/}
}
TY  - JOUR
AU  - Alvarez, Sébastien
AU  - Smith, Graham
TI  - Prescription de courbure des feuilles des laminations : retour sur un théorème de Candel
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 2549
EP  - 2593
VL  - 71
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3476/
DO  - 10.5802/aif.3476
LA  - fr
ID  - AIF_2021__71_6_2549_0
ER  - 
%0 Journal Article
%A Alvarez, Sébastien
%A Smith, Graham
%T Prescription de courbure des feuilles des laminations : retour sur un théorème de Candel
%J Annales de l'Institut Fourier
%D 2021
%P 2549-2593
%V 71
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3476/
%R 10.5802/aif.3476
%G fr
%F AIF_2021__71_6_2549_0
Alvarez, Sébastien; Smith, Graham. Prescription de courbure des feuilles des laminations : retour sur un théorème de Candel. Annales de l'Institut Fourier, Volume 71 (2021) no. 6, pp. 2549-2593. doi : 10.5802/aif.3476. https://aif.centre-mersenne.org/articles/10.5802/aif.3476/

[1] Ahlfors, Lars Lectures on quasiconformal mappings, University Lecture Series, 38, American Mathematical Society, 2006, viii+162 pages (with supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard) | MR

[2] Ahlfors, Lars; Bers, Lipman Riemann’s mapping theorem for variable metrics, Ann. Math., Volume 72 (1960), pp. 385-404 | DOI | MR | Zbl

[3] Alcalde-Cuesta, Fernando Groupoïde d’homotopie d’un feuilletage riemannien et réalisation symplectique de certaines variétés de Poisson, Publ. Mat., Barc., Volume 33 (1989) no. 3, pp. 395-410 | DOI | MR | Zbl

[4] Alcalde-Cuesta, Fernando; Dal’Bo, Françoise; Martínez, Matilde; Verjovsky, Alberto Unique ergodicity of the horocycle flow on Riemannnian foliations, Ergodic Theory Dyn. Syst., Volume 40 (2020) no. 6, pp. 1459-1479 | DOI | MR | Zbl

[5] Alcalde-Cuesta, Fernando; Hector, Gilbert Feuilletages en surfaces, cycles évanouissants et variétés de Poisson, Monatsh. Math., Volume 124 (1997) no. 3, pp. 191-213 | DOI | MR | Zbl

[6] Alvarez, Sébastien Harmonic measures and the foliated geodesic flow for foliations with negatively curved leaves, Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 2, pp. 355-374 | DOI | MR | Zbl

[7] Alvarez, Sébastien Gibbs u-states for the foliated geodesic flow and transverse invariant measures, Isr. J. Math., Volume 221 (2017) no. 2, pp. 869-940 | DOI | MR | Zbl

[8] Alvarez, Sébastien Gibbs measures for foliated bundles with negatively curved leaves, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 4, pp. 1238-1288 | DOI | MR | Zbl

[9] Alvarez, Sébastien; Brum, Joaquín Topology of leaves for minimal laminations by non-simply connected hyperbolic surfaces (2020) (https://arxiv.org/abs/2005.09050, to appear in Groups Geom. Dyn.)

[10] Alvarez, Sébastien; Brum, Joaquín; Martínez, Matilde; Potrie, Rafael Topology of leaves for minimal laminations by hyperbolic surfaces. (with an appendix with M. Wolff) (2019) (https://arxiv.org/abs/1906.10029)

[11] Alvarez, Sébastien; Lessa, Pablo The Teichmüller space of the Hirsch foliation, Ann. Inst. Fourier, Volume 68 (2018) no. 1, pp. 1-51 | DOI | Numdam | MR | Zbl

[12] Alvarez, Sébastien; Smith, Graham Earthquakes and graftings of hyperbolic surface laminations (2019) (https://arxiv.org/abs/1907.12126, to appear in Int. Math. Res. Not.)

[13] Alvarez, Sébastien; Yang, Jiagang Physical measures for the geodesic flow tangent to a transversally conformal foliation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 36 (2019) no. 1, pp. 27-51 | DOI | MR | Zbl

[14] Álvarez López, Jesús; Barral Lijó, Ramón Realization of manifolds as leaves using graph colorings (2020) (https://arxiv.org/abs/2002.08662)

[15] Álvarez López, Jesús; Barral Lijó, Ramón; Candel, Alberto A universal Riemannian foliated space, Topology Appl., Volume 198 (2016), pp. 47-85 | DOI | MR | Zbl

[16] Álvarez López, Jesús; Candel, Alberto Generic coarse geometry of leaves, Lecture Notes in Mathematics, 2223, Springer, 2018, xv+171 pages | DOI | MR

[17] Aviles, Patricio; McOwen, Robert Conformal deformations of complete manifolds with negative curvature, J. Differ. Geom., Volume 21 (1985) no. 2, pp. 269-281 | MR | Zbl

[18] Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser, 1985, vi+263 pages | DOI | MR

[19] Berger, Melvyn Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differ. Geom., Volume 5 (1971), pp. 325-332 | MR | Zbl

[20] Bland, John; Kalka, Moris Complete metrics conformal to the hyperbolic disc, Proc. Am. Math. Soc., Volume 97 (1986) no. 1, pp. 128-132 | DOI | MR | Zbl

[21] Bonatti, Christian; Gómez-Mont, Xavier; Martínez, Matilde Foliated hyperbolicity and foliations with hyperbolic leaves, Ergodic Theory Dyn. Syst., Volume 40 (2020) no. 4, pp. 881-903 | DOI | MR | Zbl

[22] Breuillard, Emmanuel; Gelander, Tsachik; Souto, Juan; Storm, Peter Dense embeddings of surface groups, Geom. Topol., Volume 10 (2006), pp. 1373-1389 | DOI | MR | Zbl

[23] Brézis, Haïm Analyse fonctionnelle. Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, 1983, xiv+234 pages | MR

[24] Calegari, Danny Foliations and the geometry of 3-manifolds, Oxford Mathematical Monographs, Oxford University Press, 2007, xiv+363 pages | MR

[25] Camacho, César; Lins Neto, Alcides Geometric theory of foliations, Birkhäuser, 1985, vi+205 pages | DOI | MR

[26] Candel, Alberto Uniformization of surface laminations, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 4, pp. 489-516 | DOI | Numdam | MR | Zbl

[27] Candel, Alberto The harmonic measures of Lucy Garnett, Adv. Math., Volume 176 (2003) no. 2, pp. 187-247 | DOI | MR | Zbl

[28] Chavel, Isaac Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, 108, Cambridge University Press, 1993, xii+386 pages | MR

[29] Cheeger, Jeff Finiteness theorems for Riemannian manifolds, Am. J. Math., Volume 92 (1970), pp. 61-74 | DOI | MR | Zbl

[30] Chow, Bennett; Knopf, Dan The Ricci flow : an introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, 2004, xii+325 pages | DOI | MR

[31] Deroin, Bertrand Nonrigidity of hyperbolic surfaces laminations, Proc. Am. Math. Soc., Volume 135 (2007) no. 3, pp. 873-881 | DOI | MR | Zbl

[32] Garnett, Lucy Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., Volume 51 (1983) no. 3, pp. 285-311 | DOI | MR | Zbl

[33] Ghys, Étienne Gauss–Bonnet theorem for 2-dimensional foliations, J. Funct. Anal., Volume 77 (1988) no. 1, pp. 51-59 | DOI | MR | Zbl

[34] Ghys, Étienne Sur l’uniformisation des laminations paraboliques, Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995) (Progress in Mathematics), Volume 145, Birkhäuser, 1997, pp. 73-91 | DOI | MR | Zbl

[35] Ghys, Étienne Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) (Panoramas et Synthèses), Volume 8, Société Mathématique de France, 1999, pp. ix, xi, 49-95 | MR | Zbl

[36] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, 224, Springer, 1983, xiii+513 pages | DOI | MR

[37] Godbillon, Claude Feuilletages : Études géométriques, Progress in Mathematics, 98, Birkhäuser, 1991, xiv+474 pages | MR

[38] Gromov, Mikhael Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, Birkhäuser, 1999, xx+585 pages (with appendices by M. Katz, P. Pansu and S. Semmes) | MR

[39] Hector, Gilbert; Hirsch, Ulrich Introduction to the geometry of foliations. Part A : Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy, Aspects of Mathematics, 1, Vieweg & Sohn, 1981, xi+234 pages | DOI | MR

[40] Hulin, Dominique; Troyanov, Marc Prescribing curvature on open surfaces, Math. Ann., Volume 293 (1992) no. 2, pp. 277-315 | DOI | MR | Zbl

[41] Kazdan, Jerry L.; Warner, Frank W. Curvature functions for compact 2-manifolds, Ann. Math., Volume 99 (1974), pp. 14-47 | DOI | MR | Zbl

[42] Kazdan, Jerry L.; Warner, Frank W. Curvature functions for open 2-manifolds, Ann. Math., Volume 99 (1974), pp. 203-219 | DOI | MR | Zbl

[43] Lessa, Pablo Reeb stability and the Gromov–Hausdorff limits of leaves in compact foliations, Asian J. Math., Volume 19 (2015) no. 3, pp. 433-463 | DOI | MR | Zbl

[44] Lessa, Pablo Brownian motion on stationary random manifolds, Stoch. Dyn., Volume 16 (2016) no. 2, 1660001, 66 pages | DOI | MR | Zbl

[45] Molino, Pierre Géométrie globale des feuilletages riemanniens, Indag. Math., New Ser., Volume 44 (1982) no. 1, pp. 45-76 | DOI | MR | Zbl

[46] Molino, Pierre Riemannian foliations, Progress in Mathematics, 73, Birkhäuser, 1988, xii+339 pages (translated from the French by Grant Cairns, with appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu) | DOI | MR

[47] Moore, Calvin C.; Schochet, Claude L. Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, 9, Cambridge University Press, 2006, xiv+293 pages | MR

[48] Muñiz, Richard; Verjovsky, Alberto Uniformization of compact foliated spaces by surfaces of hyperbolic type via the Ricci flo, Proc. Am. Math. Soc. (2021) (in Early View) | DOI | Zbl

[49] Osserman, Robert On the inequality Δuf(u), Pac. J. Math., Volume 7 (1957), pp. 1641-1647 | DOI | MR | Zbl

[50] Penner, Robert C.; Šarić, Dragomir Teichmüller theory of the punctured solenoid, Geom. Dedicata, Volume 132 (2008), pp. 179-212 | DOI | MR | Zbl

[51] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, 2006, xvi+401 pages | MR

[52] Poincaré, Henri Les fonctions fuchsiennes et l’équation Δu=e u , Journ. de Math. (5), Volume 293 (1898) no. 5, pp. 137-230

[53] Šarić, Dragomir The Teichmüller theory of the solenoid, Handbook of Teichmüller theory. Vol. II (IRMA Lectures in Mathematics and Theoretical Physics), Volume 13, European Mathematical Society, 2009, pp. 811-857 | DOI | MR | Zbl

[54] Smith, Graham Hyperbolic Plateau problems, Geom. Dedicata, Volume 176 (2015), pp. 31-44 | DOI | MR | Zbl

[55] Sullivan, Dennis Bounds, quadratic differentials, and renormalization conjectures, Mathematics into the twenty-first century. Proceedings of the AMS centennial symposium (Providence, RI, 1988) (American Mathematical Society Centennial Publications), Volume 2, American Mathematical Society, 1992, pp. 417-466 | MR | Zbl

[56] Sullivan, Dennis Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish Inc., 1993, pp. 543-564 | MR | Zbl

[57] Verjovsky, Alberto A uniformization theorem for holomorphic foliations, The Lefschetz centennial conference, Part III (Mexico City, 1984) (Contemporary Mathematics), Volume 58, American Mathematical Society, 1987, pp. 233-253 | DOI | MR | Zbl

Cited by Sources: