On the structure of Besse convex contact spheres

Mazzucchelli, Marco; Radeschi, Marco

doi:10.1090/tran/8836

On the structure of Besse convex contact spheres
HTML articles powered by AMS MathViewer

by Marco Mazzucchelli and Marco Radeschi HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 2125-2153 Request permission

Abstract:

We consider convex contact spheres $Y$ all of whose Reeb orbits are closed. Any such $Y$ admits a stratification by the periods of closed Reeb orbits. We show that $Y$ “resembles” a contact ellipsoid: any stratum of $Y$ is an integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants of $Y$ coincides with the full sequence of action values, each one repeated according to its multiplicity.

References

Alberto Abbondandolo, Barney Bramham, Umberto L. Hryniewicz, and Pedro A. S. Salomão, A systolic inequality for geodesic flows on the two-sphere, Math. Ann. 367 (2017), no. 1-2, 701–753. MR 3606452, DOI 10.1007/s00208-016-1385-2
Alberto Abbondandolo and Jungsoo Kang, Symplectic homology of convex domains and Clarke’s duality, Duke Math. J. 171 (2022), no. 3, 739–830. MR 4382979, DOI 10.1215/00127094-2021-0025
Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885, DOI 10.1007/978-3-642-61876-5
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734. MR 112160, DOI 10.2307/1970165
Daniel Cristofaro-Gardiner and Marco Mazzucchelli, The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed, Comment. Math. Helv. 95 (2020), no. 3, 461–481. MR 4152621, DOI 10.4171/CMH/493
Frank H. Clarke, A classical variational principle for periodic Hamiltonian trajectories, Proc. Amer. Math. Soc. 76 (1979), no. 1, 186–188. MR 534415, DOI 10.1090/S0002-9939-1979-0534415-7
I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys. 113 (1987), no. 3, 419–469. MR 925924, DOI 10.1007/BF01221255
I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics, Math. Z. 200 (1989), no. 3, 355–378. MR 978597, DOI 10.1007/BF01215653
Ivar Ekeland and Helmut Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z. 203 (1990), no. 4, 553–567. MR 1044064, DOI 10.1007/BF02570756
Ivar Ekeland, Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 19, Springer-Verlag, Berlin, 1990. MR 1051888, DOI 10.1007/978-3-642-74331-3
Edward R. Fadell and Paul H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), no. 2, 139–174. MR 478189, DOI 10.1007/BF01390270
Detlef Gromoll and Karsten Grove, On metrics on $S^{2}$ all of whose geodesics are closed, Invent. Math. 65 (1981/82), no. 1, 175–177. MR 636885, DOI 10.1007/BF01389300
Viktor L. Ginzburg, Başak Z. Gürel, and Marco Mazzucchelli, On the spectral characterization of Besse and Zoll Reeb flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), no. 3, 549–576. MR 4227045, DOI 10.1016/j.anihpc.2020.08.004
Hansjörg Geiges and Christian Lange, Seifert fibrations of lens spaces, Abh. Math. Semin. Univ. Hambg. 88 (2018), no. 1, 1–22. MR 3785783, DOI 10.1007/s12188-017-0188-z
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
K. Irie, Symplectic homology capacity of convex bodies and loop space homology, arXiv:1907.09749, 2019.
Samuel Lin and Benjamin Schmidt, Real projective spaces with all geodesics closed, Geom. Funct. Anal. 27 (2017), no. 3, 631–636. MR 3655958, DOI 10.1007/s00039-017-0407-x
Marco Mazzucchelli and Stefan Suhr, A characterization of Zoll Riemannian metrics on the 2-sphere, Bull. Lond. Math. Soc. 50 (2018), no. 6, 997–1006. MR 3891938, DOI 10.1112/blms.12200
Marco Mazzucchelli and Stefan Suhr, A min-max characterization of Zoll Riemannian metrics, Math. Proc. Cambridge Philos. Soc. 172 (2022), no. 3, 591–615. MR 4416570, DOI 10.1017/s0305004121000311
Christian Pries, Geodesics closed on the projective plane, Geom. Funct. Anal. 18 (2009), no. 5, 1774–1785. MR 2481742, DOI 10.1007/s00039-008-0682-7
Marco Radeschi and Burkhard Wilking, On the Berger conjecture for manifolds all of whose geodesics are closed, Invent. Math. 210 (2017), no. 3, 911–962. MR 3735632, DOI 10.1007/s00222-017-0742-4
Reinhard Schultz, Differentiability and the P. A. Smith theorems for spheres. I. Actions of prime order groups, Current trends in algebraic topology, Part 2 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 235–273. MR 686149
J.-C. Sikorav, Systèmes hamiltoniens et topologie symplectique, ETS Editrice, Pisa, 1990.
Clifford Henry Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995), no. 2, 221–238. MR 1324704, DOI 10.4310/MRL.1995.v2.n2.a10
Benjamin Texier, Basic matrix perturbation theory, Enseign. Math. 64 (2018), no. 3-4, 249–263. MR 3987143, DOI 10.4171/LEM/64-3/4-1
C. B. Thomas, Almost regular contact manifolds, J. Differential Geometry 11 (1976), no. 4, 521–533. MR 464261, DOI 10.4310/jdg/1214433722
Ilya Ustilovsky, Infinitely many contact structures on $S^{4m+1}$, Internat. Math. Res. Notices 14 (1999), 781–791. MR 1704176, DOI 10.1155/S1073792899000392
A. W. Wadsley, Geodesic foliations by circles, J. Differential Geometry 10 (1975), no. 4, 541–549. MR 400257, DOI 10.4310/jdg/1214433160