Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard

Karshon, Yael; Ziltener, Fabian

doi:10.1090/tran/7188

Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard
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by Yael Karshon and Fabian Ziltener PDF

Trans. Amer. Math. Soc. 370 (2018), 1409-1428

Abstract:

We give a complete characterization of Hamiltonian actions of compact Lie groups on exact symplectic manifolds with proper momentum maps. We deduce that every Hamiltonian action of a compact Lie group on a contractible symplectic manifold with a proper momentum map is globally linearizable.

References

Theodor Bröcker and Klaus Jänich, Introduction to differential topology, Cambridge University Press, Cambridge-New York, 1982. Translated from the German by C. B. Thomas and M. J. Thomas. MR 674117
River Chiang, Complexity one Hamiltonian $\rm SU(2)$ and $\rm SO(3)$ actions, Amer. J. Math. 127 (2005), no. 1, 129–168. MR 2115663
P. E. Conner and E. E. Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc. 10 (1959), 354–360. MR 105115, DOI 10.1090/S0002-9939-1959-0105115-X
Pierre Conner and Deane Montgomery, An example for $\textrm {SO}(3)$, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1918–1922. MR 148795, DOI 10.1073/pnas.48.11.1918
Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
Victor Guillemin and Shlomo Sternberg, A normal form for the moment map, Differential geometric methods in mathematical physics (Jerusalem, 1982) Math. Phys. Stud., vol. 6, Reidel, Dordrecht, 1984, pp. 161–175. MR 767835
Patrick Iglesias-Zemmour and Yael Karshon, Smooth Lie group actions are parametrized diffeological subgroups, Proc. Amer. Math. Soc. 140 (2012), no. 2, 731–739. MR 2846342, DOI 10.1090/S0002-9939-2011-11301-7
H. Hopf and H. Samelson, Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen, Comment. Math. Helv. 13 (1941), 240–251 (German). MR 6546, DOI 10.1007/BF01378063
Wu-chung Hsiang and Wu-yi Hsiang, Differentiable actions of compact connected classical groups. I, Amer. J. Math. 89 (1967), 705–786. MR 217213, DOI 10.2307/2373241
Yael Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Mem. Amer. Math. Soc. 141 (1999), no. 672, viii+71. MR 1612833, DOI 10.1090/memo/0672
Yael Karshon and Eugene Lerman, Non-compact symplectic toric manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 055, 37. MR 3371718, DOI 10.3842/SIGMA.2015.055
Yael Karshon and Susan Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4831–4861. MR 1852084, DOI 10.1090/S0002-9947-01-02799-4
Yael Karshon and Susan Tolman, The Gromov width of complex Grassmannians, Algebr. Geom. Topol. 5 (2005), 911–922. MR 2171798, DOI 10.2140/agt.2005.5.911
Yael Karshon and Susan Tolman, Classification of Hamiltonian torus actions with two-dimensional quotients, Geom. Topol. 18 (2014), no. 2, 669–716. MR 3180483, DOI 10.2140/gt.2014.18.669
D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), no. 4, 481–507. MR 478225, DOI 10.1002/cpa.3160310405
Friedrich Knop, Automorphisms of multiplicity free Hamiltonian manifolds, J. Amer. Math. Soc. 24 (2011), no. 2, 567–601. MR 2748401, DOI 10.1090/S0894-0347-2010-00686-8
Eugene Lerman, Eckhard Meinrenken, Sue Tolman, and Chris Woodward, Nonabelian convexity by symplectic cuts, Topology 37 (1998), no. 2, 245–259. MR 1489203, DOI 10.1016/S0040-9383(97)00030-X
John M. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics, vol. 218, Springer-Verlag, New York, 2003. MR 1930091, DOI 10.1007/978-0-387-21752-9
Ivan V. Losev, Proof of the Knop conjecture, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 1105–1134 (English, with English and French summaries). MR 2543664
Charles-Michel Marle, Modèle d’action hamiltonienne d’un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 2, 227–251 (1986) (French, with English summary). MR 859857
Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4
Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 182927, DOI 10.1090/S0002-9947-1965-0182927-5
Krzysztof Pawałowski, Fixed point sets of smooth group actions on disks and Euclidean spaces, Topology 28 (1989), no. 3, 273–289. MR 1014462, DOI 10.1016/0040-9383(89)90009-8
Reyer Sjamaar and Eugene Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375–422. MR 1127479, DOI 10.2307/2944350
Reyer Sjamaar, A de Rham theorem for symplectic quotients, Pacific J. Math. 220 (2005), no. 1, 153–166. MR 2195067, DOI 10.2140/pjm.2005.220.153
John Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481–488. MR 149457