Viterbo spectral bound conjecture on images of the zero section

Establish the existence of a constant R > 0 (depending only on the closed Riemannian manifold (M, g)) such that, for every compactly supported Hamiltonian diffeomorphism φ of T^*M satisfying φ(0_M) ⊂ DT^*M, the sheaf-theoretic spectral norm of the Lagrangian image of the zero section obeys γ(φ(0_M)) < R.

Background

The paper studies heavy and superheavy subsets in cotangent bundles via microlocal sheaf theory and connects these notions to spectral invariants and kernels quantizing Hamiltonian diffeomorphisms. A central theme is how heaviness relates to non-displaceability and to quantitative symplectic rigidity via spectral norms.

In Subsection 4.2, the author recalls a conjecture attributed to Viterbo asserting a uniform upper bound on the spectral norm of the image of the zero section under any compactly supported Hamiltonian diffeomorphism that places it inside the unit co-disc bundle. The paper shows that this conjecture would imply that every ζ_MVZ-heavy subset is ζ_MVZ-superheavy, underscoring its importance for the sheaf-theoretic approach to symplectic rigidity.