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Symplectic embeddings of $4$-dim ellipsoids into cubes

2015; Volume: 13; Issue: 4 Linguagem: Inglês

10.4310/jsg.2015.v13.n4.a2

1540-2347

Daan Frenkel, Dorothee Müller,

advanced mathematical theories

Recently, McDuff and Schlenk determined in [MS] the function c EB (a) whose value at a is the infimum of the size of a 4-ball into which the ellipsoid E(1, a) symplectically embeds (here, a 1 is the ratio of the area of the large axis to that of the smaller axis of the ellipsoid).In this paper we look at embeddings into four-dimensional cubes instead, and determine the function c EC (a) whose value at a is the infimum of the size of a 4-cube C 4 (A) = D 2 (A) × D 2 (A) into which the ellipsoid E(1, a) symplectically embeds (where D 2 (A) denotes the disc in R 2 of area A).As in the case of embeddings into balls, the structure of the graph of c EC (a) is very rich: for a less than the square σ 2 of the silver ratio σ := 1 + √ 2, the function c EC (a) turns out to be piecewise linear, with an infinite staircase converging to (σ 2 , σ 2 /2).This staircase is determined by Pell numbers.On the interval σ 2 , 7 1 32 , the function c EC (a) coincides with the volume constraint a 2 except on seven disjoint intervals, where c is piecewise linear.Finally, for a 7 1 32 , the functions c EC (a) and a 2 are equal.For the proof, we first translate the embedding problem E(1, a) → C 4 (A) to a certain ball packing problem of the ball B 4 (2A).This embedding problem is then solved by adapting the method from [MS], which finds all exceptional spheres in blow-ups of the complex projective plane that provide an embedding obstruction.We also prove that the ellipsoid E(1, a) symplectically embeds into the cube C 4 (A) if and only if E(1, a) symplectically embeds into the ellipsoid E(A, 2A).Our embedding function c EC (a) thus also describes the smallest dilate of E(1, 2) into which E(1, a) symplectically embeds.k with k ∈ {1, . . ., 7} References 1 2 1 2 3 8 9 9 10 48 49 224 225 1This result shows that, while there is symplectic rigidity for many small k, there is no rigidity at all for large k.In order to better understand these numbers, we look at a problem that interpolates the above problem of packing by k equal balls.For 0 < a 1 a 2 , consider the ellipsoid E (a 1 , a 2 ) defined above, and look for the smallest cube C(A) into which E (a 1 , a 2 ) symplectically embeds.Since E(a 1 , a 2 ) s → C(A) if and only if E(λa 1 , λa 2 ) s → C(λA), we can always assume that a 1 = 1, and therefore study the embedding capacity function c EC (a) := inf A : E(1, a) s → C(A) on the interval [1, ∞[.It is clear that c is a continuous and nondecreasing function.Since symplectic embeddings are volume preserving and the volumes of E(1, a) and C(A) are 1 2 a and A 2 respectively, we must have the lower bound a 2 c(a).It is not hard to see that k B(1)In [M2], McDuff has shown that the converse is also true!Our ellipsoid embedding problem therefore indeed interpolates the D. Frenkel and D. Müller related to the continued fraction expansion of a b .We shall explain this decomposition in more detail and prove the following proposition in the next section.Proposition 1.4.Let a, b, c, d > 0 with a b rational.Then there exists a symplectic embedding E(a, b) → P (c, d) if and only if there exists a symplectic embedding B(a, b) B(c) B(d) → B(c + d).Hutchings showed in Corollary 11 of [H2] how Proposition 1.4 implies that ECH-capacities form a complete set of invariants for the problem of symplectically embedding an ellipsoid into a polydisc: Corollary 1.5.There exists a symplectic embedding E(a, b) → P (c, d) if and only if c k ECH (E(a, b)) c k ECH (P (c, d)) for all k 0. It seems to be hard to derive Theorem 1.3 from Corollary 1.5 or vice-versa.As a further corollary we obtain Corollary 1.6.The ellipsoid E(1, a) symplectically embeds into the cube C(A) if and only if E(1, a) symplectically embeds into the ellipsoid E(A, 2A).Proof.By Corollary 1.5, E(1, a) symplectically embeds into C(A) if and only if c k ECH (E(1, a)) c k ECH (C(A)) for all k 0. By McDuff's proof of the Hofer Conjecture [M3], E(1, a) symplectically embeds into E(A, 2A) if and only if c k ECH (E(1, a)) c k ECH (E(A, 2A)) for all k 0. The corollary now follows from the remark on page 8098 in [H2], that says that for all k 0For the easy proof, we refer to Section 2.Remark 1.7.Recall that the ECH-capacities of B(1) and C(1) (or E(1, 2)) are c ECH (B(1)) = 0 ×1 , 1 ×2 , 2 ×3 , 3 ×4 , 4 ×5 , 5 ×6 , 6 ×7 , 7 ×8 , 8 ×9 , 9 ×10 , . . ., c ECH (C(1)) = 0 ×1 , 1 ×1 , 2 ×2 , 3 ×2 , 4 ×3 , 5 ×3 , 6 ×4 , 7 ×4 , 8 ×5 , 9 ×5 , . . . .One sees that the sequence c ECH (C(1)) is obtained from c ECH (B(1)) by some sort of doubling.This is reminiscent to the doubling in the definition of