Minimal entropy and geometric decompositions in dimension four

Suárez-Serrato, Pablo¹

¹CIMAT, Jalisco S/N, Col. Valenciana, CP: 36240 Guanajuato, Gto, México.

Algebraic and Geometric Topology, Tome 9 (2009) no. 1, pp. 365-395

Résumé

We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four-manifolds. We prove that any closed oriented geometric four-manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four-manifold M admits a geometric decomposition in the sense of Thurston and does not have geometric pieces modelled on hyperbolic four-space ℍ4, the complex hyperbolic plane ℍℂ2 or the product of two hyperbolic planes ℍ2 × ℍ2 then M admits an ℱ–structure. It follows that M has zero minimal entropy and collapses with curvature bounded from below. We then analyse whether or not M admits a metric whose topological entropy coincides with the minimal entropy of M and provide new examples of manifolds for which the minimal entropy problem cannot be solved.

Détail

DOI : 10.2140/agt.2009.9.365

Keywords: minimal entropy, geodesic flows, geometric structures

Bibliographie

[1] J W Anderson, G P Paternain, The minimal entropy problem for $3$–manifolds with zero simplicial volume, from: "Geometric methods in dynamics, I", Astérisque 286 (2003) 63

[2] I K Babenko, Asymptotic invariants of smooth manifolds, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) 707

[3] L Bessières, Un théorème de rigidité différentielle, Comment. Math. Helv. 73 (1998) 443

[4] G Besson, G Courtois, S Gallot, Volume et entropie minimale des espaces localement symétriques, Invent. Math. 103 (1991) 417

[5] M Brunnbauer, Homological invariance for asymptotic invariants and systolic inequalities, GAFA 18 (2008) 1087

[6] J Cheeger, M Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986) 309

[7] R J Cobb, Infra-solvmanifolds of dimension four, Bull. Austral. Math. Soc. 62 (2000) 347

[8] P Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. $(2)$ 111 (1980) 435

[9] J Eells, A Verjovsky, Harmonic and Riemannian foliations, Bol. Soc. Mat. Mexicana $(3)$ 4 (1998) 1

[10] C Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, from: "Colloque de topologie (espaces fibrés), Bruxelles, 1950", Georges Thone (1951) 29

[11] D B A Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble) 26 (1976) 265

[12] R O Filipkiewicz, Four-dimensional geometries, PhD thesis, University of Warwick (1984)

[13] W M Goldman, Complex hyperbolic geometry, Oxford Math. Monogr., Oxford Science Publ, The Clarendon Press, Oxford Univ. Press (1999)

[14] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982)

[15] P De La Harpe, Topics in geometric group theory, Chicago Lectures in Math., Univ. of Chicago Press (2000)

[16] J A Hillman, Four-manifolds, geometries and knots, Geom. Topol. Monogr. 5, Geom. Topol. Publ. (2002)

[17] M W Hirsch, Differential topology, Graduate Texts in Math. 33, Springer (1994)

[18] H Inoue, K Yano, The Gromov invariant of negatively curved manifolds, Topology 21 (1982) 83

[19] N V Ivanov, Diffeomorphism groups of Waldhausen manifolds, J. Sov. Math. 12 (1979) 115

[20] F W Kamber, P Tondeur, Flat manifolds with parallel torsion, J. Differential Geometry 2 (1968) 385

[21] R Kirby, Editor, Problems in low-dimensional topology, from: "Geometric topology (Athens, GA, 1993)", AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35

[22] S Kobayashi, K Nomizu, Foundations of differential geometry, Vol. I, Wiley Classics Library, Wiley (1996)

[23] D Kotschick, Entropies, volumes, and Einstein metrics

[24] M Kreck, W Lück, P Teichner, Counterexamples to the Kneser conjecture in dimension four, Comment. Math. Helv. 70 (1995) 423

[25] M Kreck, W Lück, P Teichner, Stable prime decompositions of four-manifolds, from: "Prospects in topology (Princeton, NJ, 1994)" (editor F Quinn), Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 251

[26] F Laudenbach, S Blank, Isotopie de formes fermées en dimension trois, Invent. Math. 54 (1979) 103

[27] C Lebrun, Kodaira dimension and the Yamabe problem, Comm. Anal. Geom. 7 (1999) 133

[28] A Manning, Topological entropy for geodesic flows, Ann. of Math. $(2)$ 110 (1979) 567

[29] W H Meeks Iii, P Scott, Finite group actions on $3$–manifolds, Invent. Math. 86 (1986) 287

[30] P Melvin, $2$–sphere bundles over compact surfaces, Proc. Amer. Math. Soc. 92 (1984) 567

[31] P Melvin, J Parker, $4$–manifolds with large symmetry groups, Topology 25 (1986) 71

[32] P Molino, Riemannian foliations, Progress in Math. 73, Birkhäuser (1988)

[33] G P Paternain, Geodesic flows, Progress in Math. 180, Birkhäuser (1999)

[34] G P Paternain, J Petean, Minimal entropy and collapsing with curvature bounded from below, Invent. Math. 151 (2003) 415

[35] G P Paternain, J Petean, Entropy and collapsing of compact complex surfaces, Proc. London Math. Soc. $(3)$ 89 (2004) 763

[36] M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Math. und ihrer Grenzgebiete 68, Springer (1972)

[37] A Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. $($N.S.$)$ 20 (1956) 47

[38] H Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. $(2)$ 77 (1963) 33

[39] M Ue, Geometric $4$–manifolds in the sense of Thurston and Seifert $4$–manifolds, I, J. Math. Soc. Japan 42 (1990) 511

[40] M Ue, Geometric $4$–manifolds in the sense of Thurston and Seifert $4$–manifolds, II, J. Math. Soc. Japan 43 (1991) 149

[41] A Verjovsky, Personal communication

[42] A W Wadsley, Geodesic foliations by circles, J. Differential Geometry 10 (1975) 541

[43] C T C Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986) 119