Abstract
We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle \(D^*M\) in a cotangent bundle \(T^*M\), when the base manifold \(M\) is an open Riemannian manifold. Our main result is that the displacement energy is not greater than \(C r(M)\), where \(r(M)\) is the inner radius of \(M\), and \(C\) is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.
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Acknowledgments
The author would like to appreciate Professor Kenji Fukaya for his warm encouragement and precious comments on the preliminary version of this paper. He also thanks the referee for many useful comments. The most part of this research was conducted when the author was supported by JSPS KAKENHI Grant No. 11J01157. The author is currently supported by JSPS KAKENHI Grant No. 25800041.
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Irie, K. Displacement energy of unit disk cotangent bundles. Math. Z. 276, 829–857 (2014). https://doi.org/10.1007/s00209-013-1224-z
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DOI: https://doi.org/10.1007/s00209-013-1224-z
Keywords
- Displacement energy
- Unit disk cotangent bundle
- Symplectic embedding problem
- Short periodic billiard trajectory
- Short geodesic loop