Many closed symplectic manifolds have infinite HoferZehnder capacity
Author:
Michael Usher
Journal:
Trans. Amer. Math. Soc. 364 (2012), 59135943
MSC (2010):
Primary 53D35, 37J45
Published electronically:
May 18, 2012
MathSciNet review:
2946937
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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 () with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal fourmanifolds having any given finitelypresented group as their fundamental group; and simply connected minimal fourmanifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature . 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 fourmanifold with admits symplectic forms with a similar property.
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Additional Information
Michael Usher
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email:
usher@math.uga.edu
DOI:
http://dx.doi.org/10.1090/S000299472012056236
PII:
S 00029947(2012)056236
Received by editor(s):
January 31, 2011
Published electronically:
May 18, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
