Topology and dynamics of laminations in surfaces of general type

Deroin, Bertrand; Dupont, Christophe

doi:10.1090/jams832

Topology and dynamics of laminations in surfaces of general type
HTML articles powered by AMS MathViewer

by Bertrand Deroin and Christophe Dupont

J. Amer. Math. Soc. 29 (2016), 495-535

DOI: https://doi.org/10.1090/jams832

Published electronically: June 2, 2015

PDF | Request permission

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

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, DOI 10.1007/BFb0103041
Thierry Barbot and Sérgio R. Fenley, Pseudo-Anosov flows in toroidal manifolds, Geom. Topol. 17 (2013), no. 4, 1877–1954. MR 3109861, DOI 10.2140/gt.2013.17.1877
David E. Barrett, Complex analytic realization of Reeb’s foliation of $S^3$, Math. Z. 203 (1990), no. 3, 355–361. MR 1038705, DOI 10.1007/BF02570743
D. E. Barrett and J. E. Fornæss, On the smoothness of Levi-foliations, Publ. Mat. 32 (1988), no. 2, 171–177. MR 975896, DOI 10.5565/PUBLMAT_{3}2288_{0}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, DOI 10.1007/978-3-642-96754-2
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, DOI 10.1017/CBO9780511623936
Marco Brunella, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 5, 569–594 (French, with English and French summaries). MR 1474805, DOI 10.1016/S0012-9593(97)89932-6
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
Marco Brunella, Courbes entières et feuilletages holomorphes, Enseign. Math. (2) 45 (1999), no. 1-2, 195–216 (French). MR 1703368
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, DOI 10.1007/s00574-007-0057-y
Danny Calegari, Foliations and the geometry of 3-manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. MR 2327361
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, DOI 10.1007/BF02698548
Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489–516. MR 1235439, DOI 10.24033/asens.1678
Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
Alberto Candel and Lawrence Conlon, Foliations. II, Graduate Studies in Mathematics, vol. 60, American Mathematical Society, Providence, RI, 2003. MR 1994394, DOI 10.1090/gsm/060
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
Bertrand Deroin, Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 1, 57–75 (French, with English and French summaries). MR 2136481, DOI 10.1016/j.ansens.2004.10.004
B. Deroin, Non rigidity of Riemann surface laminations, Proc. Amer. Math. Soc. 135 (2007), 873–881.
Bertrand Deroin and Victor Kleptsyn, Random conformal dynamical systems, Geom. Funct. Anal. 17 (2007), no. 4, 1043–1105. MR 2373011, DOI 10.1007/s00039-007-0606-y
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
J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations, Geom. Funct. Anal. 15 (2005), no. 5, 962–1003. MR 2221156, DOI 10.1007/s00039-005-0531-x
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
Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388, DOI 10.1007/978-1-4612-1688-9
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, DOI 10.1007/978-3-662-03028-8
Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Functional Analysis 51 (1983), no. 3, 285–311. MR 703080, DOI 10.1016/0022-1236(83)90015-0
Étienne Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 163–185 (1994) (French). MR 1259430, DOI 10.1007/BF02712917
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, DOI 10.1007/BF01237675
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
É. Ghys and P. de la Harpe (eds.), 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. MR 1086648, DOI 10.1007/978-1-4684-9167-8
E. Ghys and V. Sergiescu, Stabilité et conjugaison différentiable pour certains feuilletages, Topology 19 (1980), no. 2, 179–197 (French). MR 572582, DOI 10.1016/0040-9383(80)90005-1
William M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), no. 3, 557–607. MR 952283, DOI 10.1007/BF01410200
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, DOI 10.1007/BFb0061421
Michael Handel and William P. Thurston, Anosov flows on new three manifolds, Invent. Math. 59 (1980), no. 2, 95–103. MR 577356, DOI 10.1007/BF01390039
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
Sergey Ivashkovich, Vanishing cycles in holomorphic foliations by curves and foliated shells, Geom. Funct. Anal. 21 (2011), no. 1, 70–140. MR 2773104, DOI 10.1007/s00039-010-0105-4
V. A. Kaĭmanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Dokl. Akad. Nauk SSSR 288 (1986), no. 5, 1045–1049 (Russian). MR 852647
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, DOI 10.1007/BF01077429
Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613
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, DOI 10.1007/BFb0099434
François Ledrappier, Poisson boundaries of discrete groups of matrices, Israel J. Math. 50 (1985), no. 4, 319–336. MR 800190, DOI 10.1007/BF02759763
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
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, DOI 10.1007/BF02566099
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
Frank Loray and Julio C. Rebelo, Minimal, rigid foliations by curves on $\Bbb C\Bbb P^n$, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 2, 147–201. MR 1985614, DOI 10.1007/s10097-002-0049-6
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, DOI 10.1007/BFb0083068
Shigenori Matsumoto, Some remarks on foliated $S^1$ bundles, Invent. Math. 90 (1987), no. 2, 343–358. MR 910205, DOI 10.1007/BF01388709
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, DOI 10.1017/CBO9780511623813
Michael McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121–174. MR 1659270, DOI 10.1007/BF02698862
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, DOI 10.1515/9781400865178
Isao Nakai, Separatrices for nonsolvable dynamics on $\textbf {C},0$, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 569–599 (English, with English and French summaries). MR 1296744
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, DOI 10.1007/BF02679105
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
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, DOI 10.7208/chicago/9780226662237.001.0001
Ross G. Pinsky, Positive harmonic functions and diffusion, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR 1326606, DOI 10.1017/CBO9780511526244
J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), no. 2, 327–361. MR 391125, DOI 10.2307/1971034
H. Poincaré, Sur les cycles des surfaces algébriques, J. Math. Pures Appl. 5 (1902), no. 8, 169–214.
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, DOI 10.2307/2007055
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
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, DOI 10.1017/S0305004197001825
Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011
W. Thurston, Foliations on $3$-manifolds which are circle bundles (1972). Ph.D. Thesis, Berkeley.
W. Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, available at arXiv:math/9801045.
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
John W. Wood, Bundles with totally disconnected structure group, Comment. Math. Helv. 46 (1971), 257–273. MR 293655, DOI 10.1007/BF02566843