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Symplectic cohomology and a conjecture of Viterbo

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Abstract

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This settles a well-known conjecture of Viterbo from 2007 as the special case of \(T^n,\) which has been completely open for \(n>1\). Our methods are different and more intrinsic than those of the previous work of the author first settling the case \(n=1\). The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to \(C^0\) symplectic topology in particular.

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Notes

  1. This will be the case of interest in all our examples.

  2. This is false in general in the non-exact case.

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Acknowledgements

I thank Mohammed Abouzaid for suggesting to look at equivariant Lagrangians in the quantitative context, and Leonid Polterovich for motivating discussions on applying symplectic cohomology. I thank Paul Biran, Lev Buhovsky, Octav Cornea, Michael Entov, Helmut Hofer, Yanki Lekili, Alexandru Oancea, James Pascaleff, Dmitry Tonkonog, and Sara Venkatesh for useful conversations. I thank the referees for useful comments and suggestions which helped improve the exposition. This work was supported by an NSERC Discovery Grant and by the Fonds de recherche du Québec—Nature et technologies.

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  1. Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville, Montreal, QC, H3C 3J7, Canada

    Egor Shelukhin

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Shelukhin, E. Symplectic cohomology and a conjecture of Viterbo. Geom. Funct. Anal. 32, 1514–1543 (2022). https://doi.org/10.1007/s00039-022-00619-2

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