Topology and dynamics of laminations in surfaces of general type

Deroin, Bertrand¹ ; Dupont, Christophe²

¹ Ecole Normale Supérieure, DMA, UMR CNRS 8553, 45 rue d’Ulm, 75230 Paris Cedex 05, France
² Université de Rennes 1, IRMAR, UMR CNRS 6625, 35042 Rennes Cedex, France

Journal of the American Mathematical Society, Tome 29 (2016) no. 2, pp. 495-535

Résumé

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.

Détail

DOI : 10.1090/jams832

Bibliographie

[1] Ancona, A. Théorie du potentiel sur les graphes et les variétés 1990 1 112

[2] Barbot, Thierry, Fenley, Sã©Rgio R. Pseudo-Anosov flows in toroidal manifolds Geom. Topol. 2013 1877 1954

[3] Barrett, David E. Complex analytic realization of Reeb’s foliation of 𝑆³ Math. Z. 1990 355 361

[4] Barrett, D. E., Fornã¦Ss, J. E. On the smoothness of Levi-foliations Publ. Mat. 1988 171 177

[5] Barth, W., Peters, C., Van De Ven, A. Compact complex surfaces 1984

[6] Beauville, Arnaud Complex algebraic surfaces 1996

[7] Brunella, Marco Feuilletages holomorphes sur les surfaces complexes compactes Ann. Sci. École Norm. Sup. (4) 1997 569 594

[8] Brunella, Marco Birational geometry of foliations 2000 138

[9] Brunella, Marco Courbes entières et feuilletages holomorphes Enseign. Math. (2) 1999 195 216

[10] Brunella, Marco Mesures harmoniques conformes et feuilletages du plan projectif complexe Bull. Braz. Math. Soc. (N.S.) 2007 517 524

[11] Calegari, Danny Foliations and the geometry of 3-manifolds 2007

[12] Camacho, C., Lins Neto, A., Sad, P. Minimal sets of foliations on complex projective spaces Inst. Hautes Études Sci. Publ. Math. 1988

[13] Candel, Alberto Uniformization of surface laminations Ann. Sci. École Norm. Sup. (4) 1993 489 516

[14] Candel, Alberto, Conlon, Lawrence Foliations. I 2000

[15] Candel, Alberto, Conlon, Lawrence Foliations. II 2003

[16] Chavel, Isaac Eigenvalues in Riemannian geometry 1984

[17] Deroin, Bertrand Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive Ann. Sci. École Norm. Sup. (4) 2005 57 75

[18] Deroin, B. Non rigidity of Riemann surface laminations Proc. Amer. Math. Soc. 2007

[19] Deroin, Bertrand, Kleptsyn, Victor Random conformal dynamical systems Geom. Funct. Anal. 2007 1043 1105

[20] Diverio, Simone, Rousseau, Erwan A survey on hyperbolicity of projective hypersurfaces 2011

[21] Fornã¦Ss, J. E., Sibony, N. Harmonic currents of finite energy and laminations Geom. Funct. Anal. 2005 962 1003

[22] Franks, John, Williams, Bob Anomalous Anosov flows 1980 158 174

[23] Friedman, Robert Algebraic surfaces and holomorphic vector bundles 1998

[24] Friedman, Robert, Morgan, John W. Smooth four-manifolds and complex surfaces 1994

[25] Garnett, Lucy Foliations, the ergodic theorem and Brownian motion J. Functional Analysis 1983 285 311

[26] Ghys, ÉTienne Rigidité différentiable des groupes fuchsiens Inst. Hautes Études Sci. Publ. Math. 1993

[27] Ghys, ÉTienne Sur les groupes engendrés par des difféomorphismes proches de l’identité Bol. Soc. Brasil. Mat. (N.S.) 1993 137 178

[28] Ghys, ÉTienne Laminations par surfaces de Riemann 1999

[29] Sur les groupes hyperboliques d’après Mikhael Gromov 1990

[30] Ghys, E., Sergiescu, V. Stabilité et conjugaison différentiable pour certains feuilletages Topology 1980 179 197

[31] Goldman, William M. Topological components of spaces of representations Invent. Math. 1988 557 607

[32] Goodman, Sue Dehn surgery on Anosov flows 1983 300 307

[33] Handel, Michael, Thurston, William P. Anosov flows on new three manifolds Invent. Math. 1980 95 103

[34] Inaba, Takashi, Mishchenko, Michael A. On real submanifolds of Kähler manifolds foliated by complex submanifolds Proc. Japan Acad. Ser. A Math. Sci. 1994 1 2

[35] Ivashkovich, Sergey Vanishing cycles in holomorphic foliations by curves and foliated shells Geom. Funct. Anal. 2011 70 140

[36] Kaä­Manovich, V. A. Brownian motion and harmonic functions on covering manifolds. An entropic approach Dokl. Akad. Nauk SSSR 1986 1045 1049

[37] Kaä­Manovich, V. A. Brownian motion on foliations: entropy, invariant measures, mixing Funktsional. Anal. i Prilozhen. 1988 82 83

[38] Kapovich, Michael Hyperbolic manifolds and discrete groups 2001

[39] Ledrappier, F. Quelques propriétés des exposants caractéristiques 1984 305 396

[40] Ledrappier, Franã§Ois Poisson boundaries of discrete groups of matrices Israel J. Math. 1985 319 336

[41] Ledrappier, F. Profil d’entropie dans le cas continu Astérisque 1996 189 198

[42] Levitt, Gilbert Feuilletages des variétés de dimension 3 qui sont des fibres en cercles Comment. Math. Helv. 1978 572 594

[43] Loray, Frank, Marã­N Pã©Rez, David Projective structures and projective bundles over compact Riemann surfaces Astérisque 2009 223 252

[44] Loray, Frank, Rebelo, Julio C. Minimal, rigid foliations by curves on ℂℙⁿ J. Eur. Math. Soc. (JEMS) 2003 147 201

[45] Maã±Ã©, Ricardo The Hausdorff dimension of invariant probabilities of rational maps 1988 86 117

[46] Matsumoto, Shigenori Some remarks on foliated 𝑆¹ bundles Invent. Math. 1987 343 358

[47] Mattila, Pertti Geometry of sets and measures in Euclidean spaces 1995

[48] Mcquillan, Michael Diophantine approximations and foliations Inst. Hautes Études Sci. Publ. Math. 1998 121 174

[49] Mcmullen, Curtis T. Renormalization and 3-manifolds which fiber over the circle 1996

[50] Nakai, Isao Separatrices for nonsolvable dynamics on 𝐶,0 Ann. Inst. Fourier (Grenoble) 1994 569 599

[51] Nemirovskiä­, S. Yu. Stein domains with Levi-plane boundaries on compact complex surfaces Mat. Zametki 1999 632 635

[52] Ohsawa, Takeo A Levi-flat in a Kummer surface whose complement is strongly pseudoconvex Osaka J. Math. 2006 747 750

[53] Pesin, Yakov B. Dimension theory in dynamical systems 1997

[54] Pinsky, Ross G. Positive harmonic functions and diffusion 1995

[55] Plante, J. F. Foliations with measure preserving holonomy Ann. of Math. (2) 1975 327 361

[56] Poincarã©, H. Sur les cycles des surfaces algébriques J. Math. Pures Appl. 1902

[57] Reider, Igor Vector bundles of rank 2 and linear systems on algebraic surfaces Ann. of Math. (2) 1988 309 316

[58] Scott, Peter The geometries of 3-manifolds Bull. London Math. Soc. 1983 401 487

[59] Shiga, Hiroshige On monodromies of holomorphic families of Riemann surfaces and modular transformations Math. Proc. Cambridge Philos. Soc. 1997 541 549

[60] Sullivan, Dennis Cycles for the dynamical study of foliated manifolds and complex manifolds Invent. Math. 1976 225 255

[61] Thurston, W. Foliations on 3-manifolds which are circle bundles 1972

[62] Thurston, W. Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle

[63] Verjovsky, Alberto A uniformization theorem for holomorphic foliations 1987 233 253

[64] Wood, John W. Bundles with totally disconnected structure group Comment. Math. Helv. 1971 257 273