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      Many closed symplectic manifolds have infinite Hofer-Zehnder capacity

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          Abstract

          We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the K3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature 3. The examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the "neck" in the symplectic sum have no closed characteristics. We conjecture that any closed symplectic four-manifold with b^+>1 admits symplectic forms with a similar property.

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          Most cited references 13

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          Rational blowdowns of smooth 4-manifolds

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            Some simple examples of symplectic manifolds

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              The weinstein conjecture in the presence of holomorphic spheres

               H. Hofer,  C. Viterbo (1992)
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                Author and article information

                Journal
                25 January 2011
                Article
                1101.4986

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                53D35, 37J45
                Trans. Amer. Math. Soc. 364 (2012), no. 11, 5913--5943
                27 pages, 1 figure
                math.SG math.DS math.GT

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