Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Topology and dynamics of laminations in surfaces of general type


Authors: Bertrand Deroin and Christophe Dupont
Journal: J. Amer. Math. Soc. 29 (2016), 495-535
MSC (2010): Primary 32V30, 37C85; Secondary 37F75, 37C45
DOI: https://doi.org/10.1090/jams832
Published electronically: June 2, 2015
MathSciNet review: 3454381
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study Riemann surface laminations in complex algebraic surfaces of general type. We focus on the topology and dynamics of minimal sets of holomorphic foliations and on Levi-flat hypersurfaces. We begin by providing various examples. Then our first result shows that Anosov Levi-flat hypersurfaces do not embed in surfaces of general type. This allows one to classify the possible Thurston's geometries carried by Levi-flat hypersurfaces in surfaces of general type. Our second result establishes that minimal sets in surfaces of general type have a large Hausdorff dimension as soon as there exists a simply connected leaf. For both results, our methods rely on ergodic theory: we use harmonic measures and Lyapunov exponents.


References [Enhancements On Off] (What's this?)

  • [1] A. Ancona, Théorie du potentiel sur les graphes et les variétés, École d'été de Probabilités de Saint-Flour XVIII--1988, Lecture Notes in Math., vol. 1427, Springer, Berlin, 1990, pp. 1-112 (French). MR 1100282 (92g:31012), https://doi.org/10.1007/BFb0103041
  • [2] Thierry Barbot and Sérgio R. Fenley, Pseudo-Anosov flows in toroidal manifolds, Geom. Topol. 17 (2013), no. 4, 1877-1954. MR 3109861, https://doi.org/10.2140/gt.2013.17.1877
  • [3] David E. Barrett, Complex analytic realization of Reeb's foliation of $ S^3$, Math. Z. 203 (1990), no. 3, 355-361. MR 1038705 (91f:32018), https://doi.org/10.1007/BF02570743
  • [4] D. E. Barrett and J. E. Fornæss, On the smoothness of Levi-foliations, Publ. Mat. 32 (1988), no. 2, 171-177. MR 975896 (90b:32037), https://doi.org/10.5565/PUBLMAT_32288_05
  • [5] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574 (86c:32026)
  • [6] Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314 (97e:14045)
  • [7] Marco Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. Éc. Norm. Supér. (4) 30 (1997), no. 5, 569-594 (French, with English and French summaries). MR 1474805 (98i:32051), https://doi.org/10.1016/S0012-9593(97)89932-6
  • [8] Marco Brunella, Birational geometry of foliations, Monografías de Matemática. [Mathematical Monographs], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2000. Available electronically at http://www.impa.br/Publicacoes/Monografias/ Abstracts/brunella.ps. MR 1948251 (2004g:14018)
  • [9] Marco Brunella, Courbes entières et feuilletages holomorphes, Enseign. Math. (2) 45 (1999), no. 1-2, 195-216 (French). MR 1703368 (2000h:32046)
  • [10] Marco Brunella, Mesures harmoniques conformes et feuilletages du plan projectif complexe, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 4, 517-524 (French, with English and French summaries). MR 2371942 (2009c:37046), https://doi.org/10.1007/s00574-007-0057-y
  • [11] Danny Calegari, Foliations and the geometry of 3-manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. MR 2327361 (2008k:57048)
  • [12] C. Camacho, A. Lins Neto, and P. Sad, Minimal sets of foliations on complex projective spaces, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 187-203 (1989). MR 1001454 (90e:58129)
  • [13] Alberto Candel, Uniformization of surface laminations, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), no. 4, 489-516. MR 1235439 (94f:57025)
  • [14] Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868 (2002f:57058)
  • [15] Alberto Candel and Lawrence Conlon, Foliations. II, Graduate Studies in Mathematics, vol. 60, American Mathematical Society, Providence, RI, 2003. MR 1994394 (2004e:57034)
  • [16] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584 (86g:58140)
  • [17] Bertrand Deroin, Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), no. 1, 57-75 (French, with English and French summaries). MR 2136481 (2006f:32051), https://doi.org/10.1016/j.ansens.2004.10.004
  • [18] B. Deroin, Non rigidity of Riemann surface laminations, Proc. Amer. Math. Soc. 135 (2007), 873-881.
  • [19] Bertrand Deroin and Victor Kleptsyn, Random conformal dynamical systems, Geom. Funct. Anal. 17 (2007), no. 4, 1043-1105. MR 2373011 (2010j:37012), https://doi.org/10.1007/s00039-007-0606-y
  • [20] Simone Diverio and Erwan Rousseau, A survey on hyperbolicity of projective hypersurfaces, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2011. [On the title page: A survey on hiperbolicity of projective hypersurfaces]. MR 2933164
  • [21] J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations, Geom. Funct. Anal. 15 (2005), no. 5, 962-1003. MR 2221156 (2008c:32012), https://doi.org/10.1007/s00039-005-0531-x
  • [22] John Franks and Bob Williams, Anomalous Anosov flows, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 158-174. MR 591182 (82e:58078)
  • [23] Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388 (99c:14056)
  • [24] Robert Friedman and John W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994. MR 1288304 (95m:57046)
  • [25] Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), no. 3, 285-311. MR 703080 (84j:58099), https://doi.org/10.1016/0022-1236(83)90015-0
  • [26] Étienne Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 163-185 (1994) (French). MR 1259430 (95d:57009)
  • [27] Étienne Ghys, Sur les groupes engendrés par des difféomorphismes proches de l'identité, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 2, 137-178 (French, with English summary). MR 1254981 (95f:58017), https://doi.org/10.1007/BF01237675
  • [28] Étienne Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix, xi, 49-95 (French, with English and French summaries). MR 1760843 (2001g:37068)
  • [29] Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988; Edited by É. Ghys and P. de la Harpe. MR 1086648 (92f:53050)
  • [30] E. Ghys and V. Sergiescu, Stabilité et conjugaison différentiable pour certains feuilletages, Topology 19 (1980), no. 2, 179-197 (French). MR 572582 (81k:57022), https://doi.org/10.1016/0040-9383(80)90005-1
  • [31] William M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), no. 3, 557-607. MR 952283 (89m:57001), https://doi.org/10.1007/BF01410200
  • [32] Sue Goodman, Dehn surgery on Anosov flows, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 300-307. MR 1691596, https://doi.org/10.1007/BFb0061421
  • [33] Michael Handel and William P. Thurston, Anosov flows on new three manifolds, Invent. Math. 59 (1980), no. 2, 95-103. MR 577356 (81i:58032), https://doi.org/10.1007/BF01390039
  • [34] Takashi Inaba and Michael A. Mishchenko, On real submanifolds of Kähler manifolds foliated by complex submanifolds, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), no. 1, 1-2. MR 1272658 (95a:53047)
  • [35] Sergey Ivashkovich, Vanishing cycles in holomorphic foliations by curves and foliated shells, Geom. Funct. Anal. 21 (2011), no. 1, 70-140. MR 2773104 (2012g:32044), https://doi.org/10.1007/s00039-010-0105-4
  • [36] V. A. Kaĭmanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Dokl. Akad. Nauk 288 (1986), no. 5, 1045-1049 (Russian). MR 852647 (88k:58163)
  • [37] V. A. Kaĭmanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funktsional. Anal. i Prilozhen. 22 (1988), no. 4, 82-83 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 4, 326-328 (1989). MR 977003 (91b:58124), https://doi.org/10.1007/BF01077429
  • [38] Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613 (2002m:57018)
  • [39] F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d'été de probabilités de Saint-Flour, XII--1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 305-396 (French). MR 876081 (88b:58081), https://doi.org/10.1007/BFb0099434
  • [40] François Ledrappier, Poisson boundaries of discrete groups of matrices, Israel J. Math. 50 (1985), no. 4, 319-336. MR 800190 (87a:60017), https://doi.org/10.1007/BF02759763
  • [41] F. Ledrappier, Profil d'entropie dans le cas continu, Astérisque 236 (1996), 189-198 (French, with French summary). Hommage à P. A. Meyer et J. Neveu. MR 1417983 (97m:58210)
  • [42] Gilbert Levitt, Feuilletages des variétés de dimension $ 3$ qui sont des fibres en cercles, Comment. Math. Helv. 53 (1978), no. 4, 572-594 (French). MR 511848 (80c:57017), https://doi.org/10.1007/BF02566099
  • [43] Frank Loray and David Marín Pérez, Projective structures and projective bundles over compact Riemann surfaces, Astérisque 323 (2009), 223-252 (English, with English and French summaries). MR 2647972 (2011f:57031)
  • [44] Frank Loray and Julio C. Rebelo, Minimal, rigid foliations by curves on $ \mathbb{C}\mathbb{P}^n$, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 2, 147-201. MR 1985614 (2006f:37067), https://doi.org/10.1007/s10097-002-0049-6
  • [45] Ricardo Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86-117. MR 961095 (90j:58073), https://doi.org/10.1007/BFb0083068
  • [46] Shigenori Matsumoto, Some remarks on foliated $ S^1$ bundles, Invent. Math. 90 (1987), no. 2, 343-358. MR 910205 (88k:58016), https://doi.org/10.1007/BF01388709
  • [47] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890 (96h:28006)
  • [48] Michael McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121-174. MR 1659270 (99m:32028)
  • [49] Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347 (97f:57022)
  • [50] Isao Nakai, Separatrices for nonsolvable dynamics on $ {\bf C},0$, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 569-599 (English, with English and French summaries). MR 1296744 (95j:58124)
  • [51] S. Yu. Nemirovskiĭ, Stein domains with Levi-plane boundaries on compact complex surfaces, Mat. Zametki 66 (1999), no. 4, 632-635 (Russian); English transl., Math. Notes 66 (1999), no. 3-4, 522-525 (2000). MR 1747093 (2001d:32025), https://doi.org/10.1007/BF02679105
  • [52] Takeo Ohsawa, A Levi-flat in a Kummer surface whose complement is strongly pseudoconvex, Osaka J. Math. 43 (2006), no. 4, 747-750. MR 2303547 (2008a:32037)
  • [53] Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237 (99b:58003)
  • [54] Ross G. Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR 1326606 (96m:60179)
  • [55] J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), no. 2, 327-361. MR 0391125 (52 #11947)
  • [56] H. Poincaré, Sur les cycles des surfaces algébriques, J. Math. Pures Appl. 5 (1902), no. 8, 169-214.
  • [57] Igor Reider, Vector bundles of rank $ 2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309-316. MR 932299 (89e:14038), https://doi.org/10.2307/2007055
  • [58] Peter Scott, The geometries of $ 3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401-487. MR 705527 (84m:57009), https://doi.org/10.1112/blms/15.5.401
  • [59] Hiroshige Shiga, On monodromies of holomorphic families of Riemann surfaces and modular transformations, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 3, 541-549. MR 1466656 (98k:32030), https://doi.org/10.1017/S0305004197001825
  • [60] Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225-255. MR 0433464 (55 #6440)
  • [61] W. Thurston, Foliations on $ 3$-manifolds which are circle bundles (1972). Ph.D. Thesis, Berkeley.
  • [62] W. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, available at arXiv:math/9801045.
  • [63] Alberto Verjovsky, A uniformization theorem for holomorphic foliations, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 233-253. MR 893869 (88h:57027)
  • [64] John W. Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971), 257-273. MR 0293655 (45 #2732)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 32V30, 37C85, 37F75, 37C45

Retrieve articles in all journals with MSC (2010): 32V30, 37C85, 37F75, 37C45


Additional Information

Bertrand Deroin
Affiliation: Ecole Normale Supérieure, DMA, UMR CNRS 8553, 45 rue d’Ulm, 75230 Paris Cedex 05, France
Email: bertrand.deroin@ens.fr

Christophe Dupont
Affiliation: Université de Rennes 1, IRMAR, UMR CNRS 6625, 35042 Rennes Cedex, France
Email: christophe.dupont@univ-rennes1.fr

DOI: https://doi.org/10.1090/jams832
Keywords: Riemann surfaces laminations, minimal set, Levi-flat hypersurface, harmonic measure, Lyapunov exponent, surface of general type, Thurston's geometry
Received by editor(s): May 11, 2012
Received by editor(s) in revised form: July 31, 2013, November 28, 2014, and March 9, 2015
Published electronically: June 2, 2015
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement No. FP7-246918, and from the ANR projects Dynacomplexe ANR-07-JCJC-0006 and LAMBDA, ANR-13-BS01-0002.
The first author was supported by the ANR projects 08-JCJC-0130-01 and 09-BLAN-0116.
The second author was supported by the ANR project 07-JCJC-0006-01.
Dedicated: To the memory of Marco Brunella
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society