The Hofer conjecture on embedding symplectic ellipsoids. (English) Zbl 1239.53109

The necessary and sufficient conditions for symplectic embedding of ellipsoids are established by reducing the ellipsoidal embedding problem to a ball embedding problem. The embedded contact homology (ECH) methods are used by defining some quantities called ECH capacities, which are monotone under symplectic embeddings. It is shown that one open 4-dimensional ellipsoid embeds symplectically into another if and only if the ECH capacities of the first are not larger than those of the second. This proves a conjecture due to Hofer on the existence condition of the embeddings. The higher dimensional analog of this result is completely open. The results of this paper are proven by using some elementary combinatorics that develop some ideas of M. Hutchings [“Quantitative embedded contact homology”, arXiv:1005.2260], as well as the result of D. McDuff [J. Topol. 2, No. 1, 1–22 (2009; Zbl 1166.53051)].


53D35 Global theory of symplectic and contact manifolds
57R40 Embeddings in differential topology


Zbl 1166.53051
Full Text: DOI arXiv Euclid