Families of Legendrians and Lagrangians with unbounded spectral norm (Q6601740)

Spectral invariants were introduced in \textit{C. Viterbo}'s seminal work [Math. Ann. 292, No. 4, 685--710 (1992; Zbl 0735.58019)]. Spectral invariants in the symplectic case consists of functions from the group of Hamiltonian diffeomorphisms\N\[\Nc:\mathrm{Ham}\left( X,\omega\right) \rightarrow\mathbb{R}\N\]\Nthat take values in the real numbers, abiding a list of axioms.\N\N\textit{C. Viterbo} [J. Éc. Polytech., Math. 10, 67--140 (2023; Zbl 1515.37057)] conjectured that the spectral norm\N\[\N\gamma\left( CF\left( 0_{\mathbb{T}^{n}},\phi\left( 0_{\mathbb{T} ^{n}}\right) \right) \right)\N\]\Nof the Floer complex of the zero section\N\[\N0_{\mathbb{T}^{n}}\subset T^{\ast}\mathbb{T}^{n}\N\]\Nabides by a uniform bound whenever\N\[\N\phi\in \mathrm{Ham}\left( T^{\ast}\mathbb{T}^{n}\right)\N\]\Nmaps the zero section\N\[\N\phi\left( 0_{\mathbb{T}^{n}}\right) \subset DT^{\ast}\mathbb{T}^{n}\N\]\Ninto the unit-disc cotangent bundle. Recently \textit{E. Shelukhin} [Invent. Math. 230, No. 1, 321--373 (2022; Zbl 1547.53096); Geom. Funct. Anal. 32, No. 6, 1514--1543 (2022; Zbl 1509.53089)] has shown that the conjecture is valid, even for a wide range of cotangent bundles beyond the torus case. This paper gives examples of geometric settings beyond symplectic co-disc bundles, where the analogous boundedness of the spectral norm fails.\N\NThe main results are as follows.\N\N\begin{itemize}\N\item[(1)] The author shows that the spectral norm of Legendrians inside the contactization\N\[\ND^{\ast}S^{1}\times\mathbb{R}\subset J^{1}S^{1},\N\]\Nwhich are Legendrian isotopic to the zero section, does not satisfy a uniform bound.\N\N\item[(2)] \textit{P. Biran} and \textit{O. Cornea} [Comment. Math. Helv. 96, No. 4, 631--691 (2021; Zbl 1497.53134)] demonstrated that a bound\N\[\N\gamma\left( CF\left( 0^{M^{{}}},L\right) \right) \leq C\N\]\Non the spectral norm of the Floer complex of a Lagrangian\N\[\NL\subset T^{\ast}M\N\]\Nwhere \(L\)\ is Hamiltonian isotopic to the zero section, implies the bound\N\[\N\beta\left( CF\left( L,T_{pt}^{\ast}M\right) \right) \leq2C\N\]\Non the boundary depth of the Floer complex of \(L\)\ and a cotangent fiber. It is shown that the analogous result cannot be generalized to Legendrian isotopies.\N\N\item[(3)] There are Legendrian isotopy classes for which different choices of augmentations always give rise to Floer complexes that are isomorphic as filtered chain complexes, which can be characterized using the invariance of the augmentation category of \textit{F. Bourgeois} and \textit{B. Chantraine} [J. Symplectic Geom. 12, No. 3, 553--583 (2014; Zbl 1308.53119)]. It is shown that these Legendian isotopy classes admit representatives for which the Chekanov-Eliashberg algebra admits a unique augmentation.\N\N\item[(4)] The author presents a Hamiltonian isotopy of a closed exact Lagrangian inside a Liouville domain for which the spectral norm becomes arbitrarily large.\N\end{itemize}\N\NFor the entire collection see [Zbl 1515.53004].