The Hofer conjecture on embedding symplectic ellipsoids

By D. Mcduff. 2010

87 citations recorded by the source corpus.

In this note we show that one open four dimensional ellipsoid embeds symplectically into another if and only the ECH capacities of the first are no larger than those of the second. This proves a conjecture due to Hofer. The argument uses the equivalence of the ellipsoidal embedding problem with a ball embedding problem that was recently established by McDuff. Its method is inspired by Hutchings' recent results on embedded contact homology (ECH) capacities but does not use them.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/01ea6bfa5d6fefafa2e3351dc559436cbe35b104

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Citations in the corpus

Symplectic embeddings into four‐dimensional concave toric domains (2013, background): “3) It was shown by McDuff [12] (see also the survey [6]) that there exists a symplectic embedding int(E(a, b)) → E(c, d) if and only if N(a, b)k N(c, d)k for all k.”
Symplectic embeddings from concave toric domains into convex ones (2014, background): “In [23], it was then shown that since ECH capacities are known to give a sharp obstruction to all four-dimensional symplectic ball packing problems of a ball, they give sharp obstructions to ellipsoid embeddings as well.”
Beyond ECH capacities (2014, background): “McDuff [17] showed that the open ellipsoid int(E(a, b)) symplectically embeds into E(c, d) if and only if Nk(a, b) ≤ Nk(c, d) for all k.”
Symplectic embedding problems, old and new (2017, background): “Besides for the relation between embedding balls and blowing up, that is due to McDuff [76], the argument only uses the ball decomposition of ellipsoids from §6 and rather elementary results on J-holomorphic curves (the Inflation Lemma 7.3 and Gromov compactness), and it explains the equivalence fro”
The Gromov width of 4-dimensional tori (2011, background): “Recently, symplectic embeddings of ellipsoids into symplectic manifolds have attracted much attention, see [9, 10, 17, 35, 37, 40].”
The asymptotics of ECH capacities (2012, background): “For example, a theorem of McDuff [12], see also the survey [5], shows that ECH capacities give a sharp obstruction to symplectically embedding one four-dimensional ellipsoid into another.”