Viterbo spectral bound conjecture for cotangent bundles

Establish the existence of a constant R > 0 such that, for every closed Riemannian manifold (M, g) and every compactly supported Hamiltonian diffeomorphism φ ∈ Ham_c(T^*M) satisfying φ(0_M) ⊂ DT^*M, the sheaf-theoretic spectral norm of the image of the zero section obeys γ(φ(0_M)) < R. Here 0_M denotes the zero section of T^*M, DT^*M denotes the unit disk bundle determined by g, and γ(φ(0_M)) is the spectral norm associated to the Lagrangian φ(0_M).

Background

The paper introduces partial symplectic quasi-states on cotangent bundles via microlocal sheaf theory and studies heaviness/superheaviness criteria. A central theme is the relation between heaviness/superheaviness and spectral bounds.

The conjecture attributed to Viterbo asks for a uniform upper bound on the spectral norm of the image of the zero section under compactly supported Hamiltonian diffeomorphisms, provided that this image lies inside the unit disk cotangent bundle. The spectral norm γ used here is the sheaf-theoretic norm, which is shown to coincide with a Floer-theoretic norm under suitable conditions. Establishing this bound has significant consequences for the equivalence of heaviness and superheaviness in the paper.