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Israel Journal of Mathematics

, Volume 229, Issue 1, pp 39–65 | Cite as

Maximum principles in symplectic homology

  • Will J. MerryEmail author
  • Igor Uljarevic
Article
  • 42 Downloads

Abstract

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain and prove existence results for translated points.

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References

  1. [1]
    A. Abbondandolo and M. Schwarz, On the Floer homology of cotangent bundles, Communications on Pure and Applied Mathematics 59 (2006), 254–316.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz–Floer homology, Journal of Topology and Analysis 1 (2009), 307–405.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Abouzaid, Symplectic cohomology and Viterbo’s theorem, in Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 24, European Mathematical Society, Zürich, 2015, pp. 271–485.Google Scholar
  4. [4]
    P. Albers and U. Frauenfelder, Spectral invariants in Rabinowitz–Floer homology and global Hamiltonian perturbations, Journal of Modern Dynamics 4 (2010), 329–357.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    P. Albers and W. J. Merry, Translated points and Rabinowitz–Floer homology, Journal of Fixed Point Theory and Applications 13 (2013), 201–214.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    K. Barth, H. Geiges and K. Zehmisch, The diffeomorphism type of symplectic fillings, Journal of Symplectic Geometry, to appear. arXiv:1607.03310.Google Scholar
  7. [7]
    P. Biran and E. Giroux, Symplectic mapping classes and fillings, unpublished manuscript (2005).Google Scholar
  8. [8]
    K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz–Floer homology and symplectic homology, Annales Scientifiques de l’école Normale Supérieure 43 (2010), 957–1015.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Fauck, Rabinowitz–Floer homology on Brieskorn manifolds, PhD. thesis, H.U. Berlin, 2016.zbMATHGoogle Scholar
  10. [10]
    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1983.Google Scholar
  11. [11]
    Y. Groman, Floer theory on open manifolds, arXiv:1510.04265.Google Scholar
  12. [12]
    H. Hofer and D. Salamon, Floer homology and Novikov rings, in The Floer Memorial Volume, Progress in Mathematics, Vol. 133, Birkhäuser, Basel, 1995, pp. 483–524.CrossRefGoogle Scholar
  13. [13]
    J. Moser, A fixed point theorem in symplectic geometry, Acta Mathematica 141 (1978), 17–34.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R. Chiang, F. Ding and O. van Koert, Open books for Boothby–Wang bundles, fibered Dehn twists and the mean Euler characteristic, Journal of Symplectic Geometry 14 (2014), 379–426.CrossRefzbMATHGoogle Scholar
  15. [15]
    R. Chiang, F. Ding and O. van Koert, Non-fillable invariant contact structures on principal circle bundles and left-handed Dehn twists, International Journal of Mathematics 27 (2016), 55p.CrossRefzbMATHGoogle Scholar
  16. [16]
    A. F. Ritter, Deformations of symplectic cohomology and exact Lagrangians in ALE spaces, Geometric and Functional Analysis 20 (2010), 779–816.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. F. Ritter, Topological quantum field theory structure on symplectic cohomology, Journal of Topology 6 (2013), 391–489.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. F. Ritter, Floer theory for negative line bundles via Gromov–Witten invariants, Advances in Mathematics 262 (2014), 1035–1106.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. F. Ritter, Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties, Geometry & Topology 20 (2016), 1941–2052.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, UT, 1997), IAS/Park City Mathematics Series, Vol. 7, American Mathematical Society, Providence, RI, 1999, pp. 143–225.Google Scholar
  21. [21]
    D. Salamon and J. Weber, Floer homology and the heat flow, Geometric and Functional Analysis 16 (2006), 1050–1138.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S. Sandon, On iterated translated points for contactomorphisms of R2n+1 and R2n ×S1, International Journal of Mathematics 23 (2012), 14 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Seidel, p1 of symplectic automorphism groups and invertibles in quantum homology rings, Geometric and Functional Analysis 7 (1997), 1046–1095.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P. Seidel, A biased view of symplectic cohomology, in Current Developments in Mathematics, 2006, International Press, Somerville, MA, 2008, pp. 211–253.Google Scholar
  25. [25]
    P. Seidel, Exotic iterated Dehn twists, Algebraic & Geometric Topology 14 (2014), 3305–3324.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    I. Uljarevic, Floer homology of automorphisms of Liouville domains, Journal of Symplectic Geometry 15 (2017), 861–903.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    C. Viterbo, Functors and computations in Floer homology with applications. Part II, https://doi.org/front.math.ucdavis.edu/1805.01316.
  28. [28]
    C. Viterbo, Functors and computations in Floer homology with applications, I, Geometric and Functional Analysis 9 (1999), 985–1033.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZürichSwitzerland
  2. 2.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

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