Convexity package for momentum maps on contact manifolds

Chiang, River¹ ; Karshon, Yael²

¹Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
²Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 925-977

Résumé

Let a torus T act effectively on a compact connected cooriented contact manifold, and let Ψ be the natural momentum map on the symplectization. We prove that, if dimT is bigger than 2, the union of the origin with the image of Ψ is a convex polyhedral cone, the nonzero level sets of Ψ are connected (while the zero level set can be disconnected), and the momentum map is open as a map to its image. This answers a question posed by Eugene Lerman, who proved similar results when the zero level set is empty. We also analyze examples with dimT ≤ 2.

Détail

DOI : 10.2140/agt.2010.10.925

Keywords: momentum map, contact manifold, torus action, convexity

Bibliographie

[1] C Albert, Le théorème de réduction de Marsden–Weinstein en géométrie cosymplectique et de contact, J. Geom. Phys. 6 (1989) 627

[2] M F Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982) 1

[3] A Banyaga, P Molino, Géométrie des formes de contact complètement intégrables de type toriques, from: "Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1991–1992 (Montpellier)", Univ. Montpellier II (1993) 1

[4] A Barvinok, A course in convexity, Graduate Studies in Math. 54, Amer. Math. Soc. (2002)

[5] M Berger, Geometry. I, Universitext, Springer (1987)

[6] M Berger, Geometry. II, Universitext, Springer (1987)

[7] P Birtea, J P Ortega, T S Ratiu, A local-to-global principle for convexity in metric spaces, J. Lie Theory 18 (2008) 445

[8] P Birtea, J P Ortega, T S Ratiu, Openness and convexity for momentum maps, Trans. Amer. Math. Soc. 361 (2009) 603

[9] C Bjorndahl, Y Karshon, Revisiting Tietze–Nakajima – local and global convexity for maps, to appear in the Canad. J. Math.

[10] C P Boyer, K Galicki, A note on toric contact geometry, J. Geom. Phys. 35 (2000) 288

[11] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999)

[12] M Condevaux, P Dazord, P Molino, Géométrie du moment, from: "Travaux du Séminaire Sud-Rhodanien de Géométrie, I", Publ. Dép. Math. Nouvelle Sér. B 88, Univ. Claude-Bernard (1988) 131

[13] L Danzer, B Grünbaum, V Klee, Helly's theorem and its relatives, from: "Proc. Sympos. Pure Math., Vol. VII", Amer. Math. Soc. (1963) 101

[14] H Geiges, Constructions of contact manifolds, Math. Proc. Cambridge Philos. Soc. 121 (1997) 455

[15] H Geiges, An introduction to contact topology, Cambridge Studies in Advanced Math. 109, Cambridge Univ. Press (2008)

[16] V Guillemin, R Sjamaar, Convexity properties of Hamiltonian group actions, CRM Monogr. Ser. 26, Amer. Math. Soc. (2005)

[17] V Guillemin, S Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982) 491

[18] V Guillemin, S Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982) 344

[19] J Hilgert, K H Neeb, W Plank, Symplectic convexity theorems, Sem. Sophus Lie 3 (1993) 123

[20] J Hilgert, K H Neeb, W Plank, Symplectic convexity theorems and coadjoint orbits, Compositio Math. 94 (1994) 129

[21] Y Kamishima, T Tsuboi, CR-structures on Seifert manifolds, Invent. Math. 104 (1991) 149

[22] F Knop, Convexity of Hamiltonian manifolds, J. Lie Theory 12 (2002) 571

[23] E Lerman, Contact cuts, Israel J. Math. 124 (2001) 77

[24] E Lerman, A convexity theorem for torus actions on contact manifolds, Illinois J. Math. 46 (2002) 171

[25] E Lerman, Contact toric manifolds, J. Symplectic Geom. 1 (2003) 785

[26] E Lerman, E Meinrenken, S Tolman, C Woodward, Nonabelian convexity by symplectic cuts, Topology 37 (1998) 245

[27] E Lerman, C Willett, The topological structure of contact and symplectic quotients, Internat. Math. Res. Notices (2001) 33

[28] F Loose, Reduction in contact geometry, J. Lie Theory 11 (2001) 9

[29] S F B De Moraes, C Tomei, Moment maps on symplectic cones, Pacific J. Math. 181 (1997) 357

[30] S Nakajima, Über konvexe Kurven und Flächen, Tohoku Math. J. 29 (1928) 227

[31] H Nozawa, Five dimensional $K$–contact manifolds of rank 2

[32] E Prato, Convexity properties of the moment map for certain non-compact manifolds, Comm. Anal. Geom. 2 (1994) 267

[33] C V Robinson, Spherical theorems of Helly type and congruence indices of spherical caps, Amer. J. Math. 64 (1942) 260

[34] R Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. 138 (1998) 46

[35] H Tietze, Über Konvexheit im kleinen und im gro\ss en und über gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928) 697

[36] C Willett, Contact reduction, Trans. Amer. Math. Soc. 354 (2002) 4245