Celebratio Mathematica

D. McDuff: “Blow ups and symplectic embeddings in dimension 4,” Topology 30 : 3 (November 1991), pp. 409–421. MR 1113685 Zbl 0731.53035 article

D. McDuff and L. Polterovich: “Symplectic packings and algebraic geometry,” Invent. Math. 115 : 1 (1994), pp. 405–429. With an appendix by Yael Karshon. MR 1262938 Zbl 0833.53028 article

Leonid V. Polterovich	Related
Yael Karshon	Related

F. Lalonde and D. McDuff: “The geometry of symplectic energy,” Ann. Math. (2) 141 : 2 (March 1995), pp. 349–371. MR 1324138 Zbl 0829.53025 article

F. Lalonde and D. McDuff: “The classification of ruled symplectic 4-manifolds,” Math. Res. Lett. 3 : 6 (1996), pp. 769–778. MR 1426534 Zbl 0874.57019 article

D. McDuff: “From symplectic deformation to isotopy,” pp. 85–99 in Topics in symplectic 4-manifolds (Irvine, CA, 28–30 March 1996). Edited by R. J. Stern. First International Press Lecture Series 1. International Press (Cambridge, MA), 1998. MR 1635697 Zbl 0928.57018 ArXiv dg-ga/9606004 incollection

D. McDuff: “Symplectic embeddings of 4-dimensional ellipsoids,” J. Topol. 2 : 1 (2009), pp. 1–22. MR 2499436 Zbl 1166.53051 article

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 \( \mathbb{C}P^2 \). 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 \( \mathbb{C}P^2 \), 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 \( k\geq 9 \), thus answering a question raised by Hofer.

D. McDuff: “The Hofer conjecture on embedding symplectic ellipsoids,” J. Diff. Geom. 88 : 3 (2011), pp. 519–532. MR 2844441 Zbl 1239.53109 article

D. McDuff and F. Schlenk: “The embedding capacity of 4-dimensional symplectic ellipsoids,” Ann. Math. (2) 175 : 3 (2012), pp. 1191–1282. MR 2912705 Zbl 1254.53111 article

This paper calculates the function \( c(a) \) whose value at \( a \) is the infimum of the size of a ball that contains a symplectic image of the ellipsoid \( E(1,a) \). (Here \( a\geq 1 \) is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of \( c(a) \) is surprisingly rich. The volume constraint implies that \( c(a) \) is always greater than or equal to the square root of \( a \), and it is not hard to see that this is equality for large \( a \). However, for \( a \) less than the fourth power \( \tau^4 \) of the golden ratio, \( c(a) \) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval \( [\tau^4,7] \) we find \( c(a)=(a+1)/3 \). For \( a\geq 7 \), the function \( c(a) \) coincides with the square root except on a finite number of intervals where it is again piecewise linear.

The embedding constraints coming from embedded contact homology give rise to another capacity function \( c^{}_{\text{ECH}} \) which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that \( c^{}_{\text{ECH}}(a)\leq c(a) \) for all \( a \). We show here that \( c^{}_{\text{ECH}}(a) \geq c(a) \) for all \( a \).

J. Latschev, D. McDuff, and F. Schlenk: “The Gromov width of 4-dimensional tori,” Geom. Topol. 17 : 5 (2013), pp. 2813–2853. MR 3190299 Zbl 1277.57024 article

Felix Schlenk	Related
Janko Latschev	Related

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Dusa McDuff

Symplectic embedding problems

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