Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

Details Obstructions to the Existence and Squeezing of Lagrangian Cobordisms Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

2008-08-08

arxiv.org/abs/0808.1274v2

Mathematics

Symplectic Geometry

30 pages, 2 figures

Scientific paper

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 $\infty$-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.

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