Symplectic embeddings of 4‐dimensional ellipsoids

By D. Mcduff. 2008

97 citations recorded by the source corpus.

We show how to reduce the problem of symplectically embedding one 4‐dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into ℂP2. For example, the problem of embedding the ellipsoid E(1, k) into a ball B is equivalent to that of embedding k disjoint equal balls into ℂP2, and so can be solved by the work of Gromov, McDuff–Polterovich, and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1, k) for k = 1, 4 and all k ⩾ 9, thus answering a question raised by Hofer.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/08709e65a05c73cbc846bc751ab7a4e904365eed

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

The Gromov width of 4-dimensional tori (2011, methodology): “The results in McDuff [35] apply to prove the following generalization of Theorem 1.”
Quantitative embedded contact homology (2010, unknown): “It is computed in [16] that f(2) = 2 and f(5) = 5/2.”
The Hofer conjecture on embedding symplectic ellipsoids (2010, methodology): “2 The proof that the weight expansion W(p, q) described above agrees with that used in [12] is given in the appendix to [14].”
Symplectic embedding problems, old and new (2017, methodology): “2: the first step is the solution of the ball packing problem ∐k i=1 B (4)(ai) s ↪→ C(4)(A) that started with [81] in 1994, and the second step is the translation of the problem E(1, a) s ↪→ C(4)(A) to a ball packing problem [79] in 2009 (cf.”
The embedding capacity of 4-dimensional symplectic ellipsoids (2009, background): “As pointed out in McDuff [15], the Ekeland–Hofer capacities of [6] (which are purely variational) do not give all obstructions.”