Bounds on the Lagrangian spectral metric in cotangent bundles

Let $N$ be a closed manifold and $U\subset T^{\ast }(N)$ 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 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.