Bounds on the Lagrangian spectral metric in cotangent bundles

TL;DR

This paper establishes an upper bound on the Lagrangian spectral metric in cotangent bundles, linking it to Floer complex boundary depths, thus providing insights into Viterbo's conjecture about spectral distances.

Contribution

It introduces a linear upper bound on the spectral metric between Lagrangians in cotangent bundles based on Floer complex boundary depths, advancing understanding of Viterbo's conjecture.

Findings

Bound on spectral distance depends linearly on Floer boundary depth

Provides new quantitative estimates for Lagrangian spectral metrics

Supports conjecture that spectral metric is bounded in cotangent bundles

Abstract

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