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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A. Abbondandolo and M. Schwarz, On the Floer homology of cotangent bundles, Communications on Pure and Applied Mathematics 59 (2006), 254–316.
A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz–Floer homology, Journal of Topology and Analysis 1 (2009), 307–405.
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.
P. Albers and U. Frauenfelder, Spectral invariants in Rabinowitz–Floer homology and global Hamiltonian perturbations, Journal of Modern Dynamics 4 (2010), 329–357.
P. Albers and W. J. Merry, Translated points and Rabinowitz–Floer homology, Journal of Fixed Point Theory and Applications 13 (2013), 201–214.
K. Barth, H. Geiges and K. Zehmisch, The diffeomorphism type of symplectic fillings, Journal of Symplectic Geometry, to appear. arXiv:1607.03310.
P. Biran and E. Giroux, Symplectic mapping classes and fillings, unpublished manuscript (2005).
K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz–Floer homology and symplectic homology, Annales Scientifiques de l’école Normale Supérieure 43 (2010), 957–1015.
A. Fauck, Rabinowitz–Floer homology on Brieskorn manifolds, PhD. thesis, H.U. Berlin, 2016.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin, 1983.
Y. Groman, Floer theory on open manifolds, arXiv:1510.04265.
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.
J. Moser, A fixed point theorem in symplectic geometry, Acta Mathematica 141 (1978), 17–34.
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.
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.
A. F. Ritter, Deformations of symplectic cohomology and exact Lagrangians in ALE spaces, Geometric and Functional Analysis 20 (2010), 779–816.
A. F. Ritter, Topological quantum field theory structure on symplectic cohomology, Journal of Topology 6 (2013), 391–489.
A. F. Ritter, Floer theory for negative line bundles via Gromov–Witten invariants, Advances in Mathematics 262 (2014), 1035–1106.
A. F. Ritter, Circle-actions, quantum cohomology, and the Fukaya category of Fano toric varieties, Geometry & Topology 20 (2016), 1941–2052.
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.
D. Salamon and J. Weber, Floer homology and the heat flow, Geometric and Functional Analysis 16 (2006), 1050–1138.
S. Sandon, On iterated translated points for contactomorphisms of R2n+1 and R2n ×S1, International Journal of Mathematics 23 (2012), 14 pp.
P. Seidel, p1 of symplectic automorphism groups and invertibles in quantum homology rings, Geometric and Functional Analysis 7 (1997), 1046–1095.
P. Seidel, A biased view of symplectic cohomology, in Current Developments in Mathematics, 2006, International Press, Somerville, MA, 2008, pp. 211–253.
P. Seidel, Exotic iterated Dehn twists, Algebraic & Geometric Topology 14 (2014), 3305–3324.
I. Uljarevic, Floer homology of automorphisms of Liouville domains, Journal of Symplectic Geometry 15 (2017), 861–903.
C. Viterbo, Functors and computations in Floer homology with applications. Part II, https://doi.org/front.math.ucdavis.edu/1805.01316.
C. Viterbo, Functors and computations in Floer homology with applications, I, Geometric and Functional Analysis 9 (1999), 985–1033.
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Merry, W.J., Uljarevic, I. Maximum principles in symplectic homology. Isr. J. Math. 229, 39–65 (2019). https://doi.org/10.1007/s11856-018-1792-z