Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

Abstract

Abstract. Capacities that provide both qualitative and quantitative obstructions to the existence of a Lagrangian cobordism between two (n − 1)-dimensional submanifolds in parallel hyperplanes of R 2n are defined using the theory of generating families. Qualitatively, these capacities show that, for example, in R 4 there is no Lagrangian cobordism between two ∞-shaped curves with a negative crossing when the lower end is "smaller". Quantitatively, when the boundary of a Lagrangian ball lies in a hyperplane of R 2n , the capacity of the boundary gives a restriction on the size of a rectangular cylinder into which the Lagrangian ball can be squeezed.

A note on the front spinning construction

“…) for L, where f − Λ − and f Λ − are in the same equivalence class (classes are defined up to stabilizations and fiber-preserving diffeomorphisms; for more details we refer to [19]). …”

“…The way to glue two cobordisms which admit tame, compactible triples of generating families is written in [19].…”

“…(iii) Note that dim(GH i (f + Λ− )) 2l − for all i, where l − is the number of Reeb chords of Λ − . This follows from the description of the critical points of the difference function, see [13,16]. Hence, for a fixed i there exists an embedded Lagrangian filling L max Λ − of Λ − which admits a tame, compatible triple of generating families (F L max Λ − , f − ∅,max , f + Λ − ,max ) such that dim(H i (L max Λ − ; Z 2 )) = dim(H n−i+1 (L max Λ − , Λ − ; Z 2 )) = dim(GH n−i (f + Λ − ,max )) dim(GH n−i (f + Λ − )) = dim(H i (L Λ− ; Z 2 )) for every embedded Lagrangian filling L Λ− of Λ − which admits a tame, compatible triple of generating families (F LΛ − , f − ∅ , f + Λ − ).…”

The minimal length of a Lagrangian cobordism between Legendrians

“…The notion of length, however, does not seem to have an analogue in the non-relative case. In fact, an interesting feature of the length is that it arises from a 1-dimensional measurement rather than the usual 2-dimensional measurements that appear in non-squeezing theorems, even in the case of Lagrangian cobordisms [39].…”

“…Our capacities are derived from a filtered version of Legendrian Contact Cohomology, linearized by an augmentation ε; these capacities are, in a sense, monotonic under Lagrangian cobordism. Note that this framework -though not the actual construction -is similar to that used in for slices of "flat-at-infinity" Lagrangians [39] and to Hutchings ECH capacities [30]. More specifically, for each linearized Legendrian Contact Cohomology class θ ∈ LCH * (Λ, ε), we define a quantity c(Λ, ε, θ) ∈ (0, +∞] that, essentially, measures the Reeb height of the class θ. Ekholm showed that a Lagrangian cobordism L from Λ − to Λ + with an augmentation ε − for Λ − induces an augmentation ε + of Λ + and a map…”