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Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

Daniel Cristofaro-Gardiner, David Frenkel and Felix Schlenk

Algebraic & Geometric Topology 17 (2017) 1189–1260

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.

symplectic embeddings, Cremona transform
Mathematical Subject Classification 2010
Primary: 53D05
Secondary: 14B05, 32S05
Received: 26 April 2016
Accepted: 12 October 2016
Published: 14 March 2017
Daniel Cristofaro-Gardiner
Mathematics Department
Harvard University
1 Oxford Street
Cambridge, MA 02138
United States
David Frenkel
Institut de Mathématiques
Université de Neuchâtel
Rue Émile-Argand 11
CH-2000 Neuchâtel
Felix Schlenk
Institut de Mathématiques
Université de Neuchâtel
Rue Émile-Argand 11
CH-2000 Neuchâtel