Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

Cristofaro-Gardiner, Daniel¹ ; Frenkel, David² ; Schlenk, Felix²

¹Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, United States
²Institut de Mathématiques, Université de Neuchâtel, Rue Émile-Argand 11, CH-2000 Neuchâtel, Switzerland

Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 1189-1260

Résumé

In previous work, the second author and Müller determined the function c(a) giving the smallest dilate of the polydisc P(1,1) into which the ellipsoid E(1,a) symplectically embeds. We determine the function of two variables cb(a) giving the smallest dilate of the polydisc P(1,b) into which the ellipsoid E(1,a) symplectically embeds for all integers b ≥ 2.

It is known that, for fixed b, if a is sufficiently large then all obstructions to the embedding problem vanish except for the volume obstruction. We find that there is another kind of change of structure that appears as one instead increases b: the number-theoretic “infinite Pell stairs” from the b = 1 case almost completely disappears (only two steps remain) but, in an appropriately rescaled limit, the function cb(a) converges as b tends to infinity to a completely regular infinite staircase with steps all of the same height and width.

Détail

DOI : 10.2140/agt.2017.17.1189

Classification : 53D05, 14B05, 32S05
Keywords: symplectic embeddings, Cremona transform

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