Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

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 ${\mathbb{R}}^{2n}$ are defined using the theory of generating families. Qualitatively, these capacities show that, for example, in ${\mathbb{R}}^4$ there is no Lagrangian cobordism between two $\mathrm{\infty }$-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 ${\mathbb{R}}^{2n}$, the capacity of the boundary gives a restriction on the size of a rectangular cylinder into which the Lagrangian ball can be squeezed.