Viterbo conjecture for Zoll symmetric spaces

2018/11/13 by Egor Shelukhin, Shelukhin, Egor · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1811.05552

openalex publication_date 2018/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces Sⁿ, ℝ Pⁿ, ℂ Pⁿ, ℍ Pⁿ, n≥ 1. We discuss generalizations and give applications, in particular to C⁰ symplectic topology. Our key methods, which are of independent interest, consist of a reinterpretation of the spectral norm via the asymptotic behavior of a family of cones of filtered morphisms, and a quantitative deformation argument for Floer persistence modules, that allows to excise a divisor.

Cited by

• Hofer-continuous quasi-morphisms on Liouville manifolds
• Persistence modules induced by inner functions
• Universal topological statistics on triangulated singular spaces
• C⁰-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

Related

• Symplectic cohomology and a conjecture of Viterbo
• Viterbo's spectral bound conjecture for homogeneous spaces
• Families of Legendrians and Lagrangians with unbounded spectral norm
• Symplectic cohomology and Viterbo's theorem
• Bounds on the Lagrangian spectral metric in cotangent bundles