×

Prescribing the curvature of leaves of laminations: revisiting a theorem by Candel. (Prescription de courbure des feuilles des laminations: retour sur un théorème de Candel.) (French. English summary) Zbl 07554454

Summary: 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.

MSC:

57R30 Foliations in differential topology; geometric theory
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
30F10 Compact Riemann surfaces and uniformization
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ahlfors, Lars, Lectures on quasiconformal mappings, 38, viii+162 p. pp. (2006), American Mathematical Society · Zbl 1103.30001
[2] Ahlfors, Lars; Bers, Lipman, Riemann’s mapping theorem for variable metrics, Ann. Math., 72, 385-404 (1960) · Zbl 0104.29902 · doi:10.2307/1970141
[3] Alcalde-Cuesta, Fernando, Groupoïde d’homotopie d’un feuilletage riemannien et réalisation symplectique de certaines variétés de Poisson, Publ. Mat., Barc., 33, 3, 395-410 (1989) · Zbl 0709.58014
[4] Alcalde-Cuesta, Fernando; Dal’Bo, Françoise; Martínez, Matilde; Verjovsky, Alberto, Unique ergodicity of the horocycle flow on Riemannnian foliations, Ergodic Theory Dyn. Syst., 40, 6, 1459-1479 (2020) · Zbl 1445.37024 · doi:10.1017/etds.2018.119
[5] Alcalde-Cuesta, Fernando; Hector, Gilbert, Feuilletages en surfaces, cycles évanouissants et variétés de Poisson, Monatsh. Math., 124, 3, 191-213 (1997) · Zbl 0888.57027 · doi:10.1007/BF01298244
[6] Alvarez, Sébastien, Harmonic measures and the foliated geodesic flow for foliations with negatively curved leaves, Ergodic Theory Dyn. Syst., 36, 2, 355-374 (2016) · Zbl 1355.37054 · doi:10.1017/etds.2014.59
[7] Alvarez, Sébastien, Gibbs \(u\)-states for the foliated geodesic flow and transverse invariant measures, Isr. J. Math., 221, 2, 869-940 (2017) · Zbl 1379.37072 · doi:10.1007/s11856-017-1578-8
[8] Alvarez, Sébastien, Gibbs measures for foliated bundles with negatively curved leaves, Ergodic Theory Dyn. Syst., 38, 4, 1238-1288 (2018) · Zbl 1388.37038 · doi:10.1017/etds.2016.76
[9] Alvarez, Sébastien; Brum, Joaquín, Topology of leaves for minimal laminations by non-simply connected hyperbolic surfaces (2020) · Zbl 1497.57041
[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)
[11] Alvarez, Sébastien; Lessa, Pablo, The Teichmüller space of the Hirsch foliation, Ann. Inst. Fourier, 68, 1, 1-51 (2018) · Zbl 1409.57029
[12] Alvarez, Sébastien; Smith, Graham, Earthquakes and graftings of hyperbolic surface laminations (2019) · Zbl 1489.57008
[13] Alvarez, Sébastien; Yang, Jiagang, Physical measures for the geodesic flow tangent to a transversally conformal foliation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 36, 1, 27-51 (2019) · Zbl 1404.53037 · doi:10.1016/j.anihpc.2018.03.009
[14] Álvarez López, Jesús; Barral Lijó, Ramón, Realization of manifolds as leaves using graph colorings (2020)
[15] Álvarez López, Jesús; Barral Lijó, Ramón; Candel, Alberto, A universal Riemannian foliated space, Topology Appl., 198, 47-85 (2016) · Zbl 1336.53055 · doi:10.1016/j.topol.2015.11.006
[16] Álvarez López, Jesús; Candel, Alberto, Generic coarse geometry of leaves, 2223, xv+171 p. pp. (2018), Springer · Zbl 1426.57001 · doi:10.1007/978-3-319-94132-5
[17] Aviles, Patricio; McOwen, Robert, Conformal deformations of complete manifolds with negative curvature, J. Differ. Geom., 21, 2, 269-281 (1985) · Zbl 0588.53028
[18] Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor, Manifolds of nonpositive curvature, 61, vi+263 p. pp. (1985), Birkhäuser · Zbl 0591.53001 · doi:10.1007/978-1-4684-9159-3
[19] Berger, Melvyn, Riemannian structures of prescribed Gaussian curvature for compact \(2\)-manifolds, J. Differ. Geom., 5, 325-332 (1971) · Zbl 0222.53042
[20] Bland, John; Kalka, Moris, Complete metrics conformal to the hyperbolic disc, Proc. Am. Math. Soc., 97, 1, 128-132 (1986) · Zbl 0592.53029 · doi:10.2307/2046093
[21] Bonatti, Christian; Gómez-Mont, Xavier; Martínez, Matilde, Foliated hyperbolicity and foliations with hyperbolic leaves, Ergodic Theory Dyn. Syst., 40, 4, 881-903 (2020) · Zbl 1439.37041 · doi:10.1017/etds.2018.61
[22] Breuillard, Emmanuel; Gelander, Tsachik; Souto, Juan; Storm, Peter, Dense embeddings of surface groups, Geom. Topol., 10, 1373-1389 (2006) · Zbl 1132.22011 · doi:10.2140/gt.2006.10.1373
[23] Brézis, Haïm, Analyse fonctionnelle. Théorie et applications, xiv+234 p. pp. (1983), Masson · Zbl 0511.46001
[24] Calegari, Danny, Foliations and the geometry of 3-manifolds, xiv+363 p. pp. (2007), Oxford University Press · Zbl 1118.57002
[25] Camacho, César; Lins Neto, Alcides, Geometric theory of foliations, vi+205 p. pp. (1985), Birkhäuser · Zbl 0568.57002 · doi:10.1007/978-1-4612-5292-4
[26] Candel, Alberto, Uniformization of surface laminations, Ann. Sci. Éc. Norm. Supér., 26, 4, 489-516 (1993) · Zbl 0785.57009
[27] Candel, Alberto, The harmonic measures of Lucy Garnett, Adv. Math., 176, 2, 187-247 (2003) · Zbl 1031.58003 · doi:10.1016/S0001-8708(02)00036-1
[28] Chavel, Isaac, Riemannian geometry—a modern introduction, 108, xii+386 p. pp. (1993), Cambridge University Press · Zbl 0810.53001
[29] Cheeger, Jeff, Finiteness theorems for Riemannian manifolds, Am. J. Math., 92, 61-74 (1970) · Zbl 0194.52902
[30] Chow, Bennett; Knopf, Dan, The Ricci flow : an introduction, 110, xii+325 p. pp. (2004), American Mathematical Society · Zbl 1086.53085 · doi:10.1090/surv/110
[31] Deroin, Bertrand, Nonrigidity of hyperbolic surfaces laminations, Proc. Am. Math. Soc., 135, 3, 873-881 (2007) · Zbl 1110.57018 · doi:10.1090/S0002-9939-06-08579-0
[32] Garnett, Lucy, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51, 3, 285-311 (1983) · Zbl 0524.58026 · doi:10.1016/0022-1236(83)90015-0
[33] Ghys, Étienne, Gauss-Bonnet theorem for \(2\)-dimensional foliations, J. Funct. Anal., 77, 1, 51-59 (1988) · Zbl 0656.57017 · doi:10.1016/0022-1236(88)90076-6
[34] Ghys, Étienne, Integrable systems and foliations/Feuilletages et systèmes intégrables (Montpellier, 1995), 145, Sur l’uniformisation des laminations paraboliques, 73-91 (1997), Birkhäuser · Zbl 0879.32024 · doi:10.1007/978-1-4612-4134-8_5
[35] Ghys, Étienne, Dynamique et géométrie complexes (Lyon, 1997), 8, Laminations par surfaces de Riemann, ix, xi, 49-95 (1999), Société Mathématique de France · Zbl 1018.37028
[36] Gilbarg, David; Trudinger, Neil S., Elliptic partial differential equations of second order, 224, xiii+513 p. pp. (1983), Springer · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0
[37] Godbillon, Claude, Feuilletages : Études géométriques, 98, xiv+474 p. pp. (1991), Birkhäuser · Zbl 0724.58002
[38] Gromov, Mikhael, Metric structures for Riemannian and non-Riemannian spaces, 152, xx+585 p. pp. (1999), Birkhäuser · Zbl 0953.53002
[39] Hector, Gilbert; Hirsch, Ulrich, Introduction to the geometry of foliations. Part A : Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy, 1, xi+234 p. pp. (1981), Vieweg & Sohn · Zbl 0486.57002
[40] Hulin, Dominique; Troyanov, Marc, Prescribing curvature on open surfaces, Math. Ann., 293, 2, 277-315 (1992) · Zbl 0799.53047 · doi:10.1007/BF01444716
[41] Kazdan, Jerry L.; Warner, Frank W., Curvature functions for compact \(2\)-manifolds, Ann. Math., 99, 14-47 (1974) · Zbl 0273.53034
[42] Kazdan, Jerry L.; Warner, Frank W., Curvature functions for open \(2\)-manifolds, Ann. Math., 99, 203-219 (1974) · Zbl 0278.53031
[43] Lessa, Pablo, Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations, Asian J. Math., 19, 3, 433-463 (2015) · Zbl 1323.57017 · doi:10.4310/AJM.2015.v19.n3.a3
[44] Lessa, Pablo, Brownian motion on stationary random manifolds, Stoch. Dyn., 16, 2, 66 p. pp. (2016) · Zbl 1335.60156 · doi:10.1142/S0219493716600017
[45] Molino, Pierre, Géométrie globale des feuilletages riemanniens, Indag. Math., New Ser., 44, 1, 45-76 (1982) · Zbl 0516.57016
[46] Molino, Pierre, Riemannian foliations, 73, xii+339 p. pp. (1988), Birkhäuser · Zbl 0633.53001 · doi:10.1007/978-1-4684-8670-4
[47] Moore, Calvin C.; Schochet, Claude L., Global analysis on foliated spaces, 9, xiv+293 p. pp. (2006), Cambridge University Press · Zbl 1091.58015
[48] Muñiz, Richard; Verjovsky, Alberto, Uniformization of compact foliated spaces by surfaces of hyperbolic type via the Ricci flo, Proc. Am. Math. Soc. (2021) · Zbl 1489.53034 · doi:10.1090/proc/15780
[49] Osserman, Robert, On the inequality \(\Delta u\ge f(u)\), Pac. J. Math., 7, 1641-1647 (1957) · Zbl 0083.09402
[50] Penner, Robert C.; Šarić, Dragomir, Teichmüller theory of the punctured solenoid, Geom. Dedicata, 132, 179-212 (2008) · Zbl 1182.30076 · doi:10.1007/s10711-007-9226-9
[51] Petersen, Peter, Riemannian geometry, 171, xvi+401 p. pp. (2006), Springer · Zbl 1220.53002
[52] Poincaré, Henri, Les fonctions fuchsiennes et l’équation \({\Delta } u=e^u\), Journ. de Math. (5), 293, 5, 137-230 (1898) · JFM 29.0367.01
[53] Šarić, Dragomir, Handbook of Teichmüller theory. Vol. II, 13, The Teichmüller theory of the solenoid, 811-857 (2009), European Mathematical Society · Zbl 1179.30042 · doi:10.4171/055-1/20
[54] Smith, Graham, Hyperbolic Plateau problems, Geom. Dedicata, 176, 31-44 (2015) · Zbl 1318.53069 · doi:10.1007/s10711-014-9958-2
[55] Sullivan, Dennis, Mathematics into the twenty-first century. Proceedings of the AMS centennial symposium (Providence, RI, 1988), 2, Bounds, quadratic differentials, and renormalization conjectures, 417-466 (1992), American Mathematical Society · Zbl 0936.37016
[56] Sullivan, Dennis, Topological methods in modern mathematics (Stony Brook, NY, 1991), Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, 543-564 (1993), Publish or Perish Inc. · Zbl 0803.58018
[57] Verjovsky, Alberto, The Lefschetz centennial conference, Part III (Mexico City, 1984), 58, A uniformization theorem for holomorphic foliations, 233-253 (1987), American Mathematical Society · Zbl 0619.32017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.