Bounds on the Lagrangian spectral metric in cotangent bundles

Let N be a closed manifold and U subset of T* (N) a bounded domain in the cotangent bundle of N, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds in U that are exact-isotopic to the zero-section is bounded. In this paper we establish an upper bound on the spectral distance between two such Lagrangians L-0, L-1, which depends linearly on the boundary depth of the Floer complexes of (L-0, F) and (L-1, F), where F is a fiber of the cotangent bundle.