Symplectic embeddings of four-dimensional ellipsoids into polydiscs

Abstract

McDuff and Schlenk have recently determined exactly when a fourdimensional symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly, Frenkel and Müller have recently determined exactly when a symplectic ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings of more complicated structures, however, remain mostly unexplored. We study when a symplectic ellipsoid E(a, b) symplectically embeds into a polydisc P (c, d). We prove that there exists a constant C depending only on d/c (here, d is assumed greater than c) such that if b/a is greater than C, then the only obstruction to symplectically embedding E(a, b) into P (c, d) is the volume obstruction. We also conjecture exactly when an ellipsoid embeds into a scaling of P (1, b) for b greater than or equal to 6, and conjecture about the set of (a, b) such that the only obstruction to embedding E(1, a) into a scaling of P (1, b) is the classical volume. Finally, we verify our conjecture for b = 13 2 .

Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs

“…The function c B (a) has three parts: On [1, τ 4 ], with τ = 1+ √ 5 2 the golden ratio, c B is given by the "Fibonacci stairs", namely an infinite stairs each of whose steps is made of a segment on a line going through the origin and a horizontal segment, with foot-points on the volume constraint √ a, and both the foot-points and the edge determined by Fibonacci numbers. Then there is one step over [τ 4 , 7 1 9 ], whose left part over [τ 4 , 7] is affine but non-linear: c B (a) = a+1 3 . Finally, for a 7 1 9 the graph of c B (a) is given by eight strictly disjoint steps made of two affine segments, and c B (a) = √ a for a 8 1 36 .…”

“…The reduction method is thus quite "local in a". While it is usually impossible to compute c b (a) by Method 3 (see however [4,14]), this method is very useful for guessing the graph of c b (a), since using (2.15) and a computer one gets good lower bounds for c b (a). Accordingly, we have found Theorem 1.1 as follows.…”

“…The smallest interesting dimension is four, and all our results are in this dimension. So consider the standard four-dimensional symplectic vector space (Ê 4 , ω), where ω = dx 1 ∧dy 1 +dx 2 ∧dy 2 . Open subsets in Ê 4 are endowed with the same symplectic form.…”

“…It would be particularly interesting to understand this film for b ∈ [1,2], or just for b ∈ [1, 1 + ε] for some ε > 0, namely to understand how the Pell stairs disappear. In [4], ECH-capacities are used to compute c b (a) for b = 13 2 and to get an idea of this film. In accordance with Theorem 1.1, Conjecture 6.3 in [4] and further investigations we make the The obstructions given by the exceptional classes E n and F n are readily computed, see § 3.3: While the classes E n again give rise to a finite staircase with linear steps, the classes F n give an obstruction only for b ∈ (n − n (n+1) 2 , n + 1 n+2 ).…”

“…In [4], ECH-capacities are used to compute c b (a) for b = 13 2 and to get an idea of this film. In accordance with Theorem 1.1, Conjecture 6.3 in [4] and further investigations we make the The obstructions given by the exceptional classes E n and F n are readily computed, see § 3.3: While the classes E n again give rise to a finite staircase with linear steps, the classes F n give an obstruction only for b ∈ (n − n (n+1) 2 , n + 1 n+2 ). While our proof of Theorem 1.1 should extend to a proof of Conjecture 1.5, the analysis is more involved, since fractional parts arise, that are harder to estimate.…”