Abstract
We prove an inequality between the \(L^{\infty }\)-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals et al. (J Symplectic Geom 14:1013–1031, 2016). As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimisers of the \(L^{\infty }\)-norm. Moreover, we establish \(L^\infty \)-type contact systolic inequalities in the presence of a positive loop.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbondandolo, A., Bramham, B., Hryniewicz, U., Salomao, P.A.: Sharp systolic inequalities for Reeb flows on the three-sphere. arXiv:1504.0525 (2015)
Albers, P., Frauenfelder, U.: Leaf-wise intersections and Rabinowitz Floer homology. J. Topol. Anal. 2(1), 77–98 (2010)
Albers, P., Frauenfelder, U.: Rabinowitz Floer homology: A survey, global differential geometry, vol. 17. In: Springer Proceedings in Mathematics, no. 3, Springer, pp. 437–461 (2012)
Albers, P., Frauenfelder, U.: A variational approach to Givental’s nonlinear Maslov index. GAFA 22(5), 1033–1050 (2012)
Albers, P., Fuchs, U., Merry, W.J.: Orderability and the Weinstein conjecture. Compos. Math. 151, 2251–2272 (2015)
Albers, P., Merry, W.J.: Translated points and Rabinowitz Floer homology. J. Fixed Point Theory Appl. 13(1), 201–214 (2013)
Alvarez Paiva, J.C., Balacheff, F.: Contact geometry and isosystolic inequalities. Geom. Funct. Anal. 24, 648–669 (2014)
Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)
Bourgeois, F., Oancea, A.: Fredholm theory and transversality for the parametrized and for the \(S^{1}\)-invariant symplectic action. J. Eur. Math. Soc. 12(5), 1181–1229 (2010)
Cieliebak, K., Frauenfelder, U.: A Floer homology for exact contact embeddings. Pac. J. Math. 239(2), 216–251 (2009)
Colin, V., Honda, K.: Constructions contrôlées de champs de Reeb et applications. Geom. Topol. 9, 2193–2226 (2005)
Cieliebak, K., Mohnke, K.: Compactness for punctured holomorphic curves. J. Symplectic Geom. 3(4), 589–654 (2005)
Casals, R., Presas, F.: On the strong orderability of overtwisted 3-folds. Comment. Math. Helv. 91, 305–316 (2016)
Casals, R., Presas, F., Sandon, S.: On the non-existence of small positive loops of contactomorphisms on overtwisted contact manifolds. J. Symplectic Geom. 14, 1013–1031 (2016)
Eliashberg, Y., Kim, S.S., Polterovich, L.: Geometry of contact transformations and domains: orderability versus squeezing. Geom. Topol. 10, 1635–1747 (2006)
Eliashberg, Y., Polterovich, L.: Partially ordered groups and geometry of contact transformations. Geom. Funct. Anal. 10(6), 1448–1476 (2000)
Fish, J.: Target-local Gromov compactness. Geom. Topol. 15(2), 765–826 (2011)
Frauenfelder, U., Labrousse, C., Schlenk, F.: Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems. J. Topol. Anal. 7(3), 407–451 (2015)
Frauenfelder, U.: The Arnold–Givental conjecture and moment Floer homology. Int. Math. Res. Not. 42, 2179–2269 (2004)
Gutt, J.: The positive equivariant symplectic homology as an invariant for some contact manifolds. arXiv:1503.1443 (2015)
Hofer, H., Wysocki, C., Zehnder, E.: A general Fredholm theory II: implicit function theorems. Geom. Funct. Anal. 19, 206–293 (2009)
Hofer, H., Wysocki, K., Zehnder, E.: Polyfold and Fredholm Theory I: basic theory in M-polyfolds. arXiv:1407.3185 (2014)
Rabinowitz, P.: Periodic solutions of a Hamiltonian system on a prescribed energy surface. J. Differ. Equ. 33(3), 336–352 (1979)
Salamon, D.: In: Eliashberg, Y., Traynor, L. (eds.) Lectures on Floer Homology, Symplectic Geometry and Topology, IAS/Park City Math. Series, vol. 7, Am. Math. Soc., pp. 143–225 (1999)
Sandon, S.: Floer homology for translated points (2016)
Sandon, S.: A Morse estimate for translated points of contactomorphisms of spheres and projective spaces. Geom. Dedic. 165, 95–110 (2013)
Shelukhin, E.: The Hofer norm of a contactomorphism. arXiv:1411.1457 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Albers, P., Fuchs, U. & Merry, W.J. Positive loops and \(L^{\infty }\)-contact systolic inequalities. Sel. Math. New Ser. 23, 2491–2521 (2017). https://doi.org/10.1007/s00029-017-0338-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-017-0338-2