Lagrangian cobordism and metric invariants

Abstract

We introduce new pseudo-metrics on spaces of Lagrangian submanifolds of a symplectic manifold (M, ω) by considering areas associated to projecting Lagrangian cobordisms in C×M to the "time-energy plane" C. We investigate the non-degeneracy properties of these pseudo-metrics, reflecting the rigidity and flexibility aspects of Lagrangian cobordisms.

Bounds on spectral norms and barcodes

“…First, following Chekanov's argument for his non-displacement theorem [29] (see also [11,26,33]) one has the following consequence of a Hamiltonian H ∈ H having small spectral norm. Whenever it applies, this statement sharpens the previous results in this direction by Inequality (4).…”

“…In this setting the following sharper version of Chekanov's non-degeneracy theorem [30] (see also [11,26,33]) holds. It is indeed a sharpening by Inequality 4.…”

“…The interleaving bounds, as well as the bound on A(z), follow from standard action estimates (cf. [33,Lemma 3.11] and references therein).…”

“…Non-displacement. In this section we prove Theorem D, using notions from Section 4 and the approach in [11,26,29,33]. Since these methods are by now quite standard, we outline the main points, leaving the details to the interested reader.…”

“…We continue by observing that given F ∈ H and J ∈ J M , for each interval [v, w) of length 0 < w − v < = (J , L), with v, w / ∈ Spec(L, F ) ∪ Spec(L, 0) the Floer complexes CF (L, 0; D) [v,w) , CF (L, F ; D) [v,w) generated by capped orbits in O η (L, F ; D) with actions in [v, w) are well-defined, for perturbation data D with Hamiltonian term sufficiently C 2 -small, and almost complex structure J D being a small perturbation of J . This follows by Gromov compactness, and associated standard bubbling analysis (see [33,Section 3.5]). We denote their homologies by HF (L, 0; D) [v,w) , HF (L, F ; D) [v,w) .…”