Families of Legendrians and Lagrangians with unbounded spectral norm

TL;DR

This paper investigates the limitations of spectral norm bounds for Legendrians and Lagrangians in extended geometric settings, showing that such bounds can fail under certain natural generalizations.

Contribution

It identifies specific scenarios where the spectral norm bound, conjectured for Lagrangians in a torus, does not hold, extending understanding of spectral invariants in contact and Weinstein settings.

Findings

Spectral norm bounds fail for Legendrian isotopies in contactisation.

Spectral norm bounds fail after attaching Weinstein one-handles.

The results highlight limitations of spectral invariants in generalized geometric contexts.

Abstract

Viterbo has conjectured that any Lagrangian in the unit co-disc bundle of a torus which is Hamiltonian isotopic to the zero-section satisfies a uniform bound on its spectral norm; a recent result by Shelukhin showed that this is indeed the case. The modest goal of this note is to explore two natural generalisations of the above geometric setting in which the bound of the spectral norm fails: first, passing to Legendrian isotopies in the contactisation of the unit co-disc bundle (Hamiltonian isotopies lift to Legendrian isotopies) and, second, considering Hamiltonian isotopies but after modifying the co-disc bundle by attaching a critical Weinstein one-handle.