Abstract
This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has ‘eccentricity’ of the form \(3\ell -1\) for some integer \(\ell \).
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Notes
The capacity, being an area, is a much more natural measure in symplectic geometry than the radius.
See [3] for further discussion of the significance of this function.
The symplectic gradient vector field \(X_H\) is given by the identity
, and is tangent to the kernel of \(\omega |_{H=\mathrm{const}}.\)
Strictly speaking, by “curve” we mean an equivalence class of maps
, where we quotient out by the finite dimensional group of biholomorphisms of the domain. In the situations considered here, this group is trivial whenever the total number of punctures or marked points on the domain is at least three.See [3, Section2] for more details of these calculations. Further, the degree of C is the number of times it intersects the line at infinity.
A negative end is modelled on
, while a positive end is modelled on 
. Thus negative ends are concave (contracting as one moves towards \(-\infty \)), while positive ends are convex (expanding towards \(+\infty \)).By definition, a trivial cylinder in a symplectization
is an unbranched multiple cover of the cylinder 
over an asymptotic Reeb orbit \(\beta \) in \({\mathrm{\Sigma }}\). One might be able to weaken this requirement, but to do this would require detailed analysis of the gluing argument. The point is that in this case the argument is not local to the top end of u, but because of its invariance under the \({\mathbb R}\)-action, involves a neighborhood of the whole curve.The precise value of \(\lambda \) is irrelevant, since all the curves we construct here persist under deformations.
This somewhat surprising interpretation of the ECH index holds because the latter is a sum of Conley–Zehnder Fredholm-type indices that as above have the form
and hence are related to numbers of integer points: see [11].Despite appearances, this example is not completely random. Indeed, \( {76}/{11} = 7- 1/{11}\) bears the same relation to 11 as the Fibonacci quotient 55 / 8 does to 8: i.e. in the notation of [15, Lemma 4.1.2] we have \({76}/{11}= v_1(10)\) while \({55}/{8}= v_1(7)\). Thus 76 / 11 is the first in the next set of generalized ghost stairs.
Note that we cannot assert the existence of such a building by means of a cobordism argument in cylindrical contact homology because the top end is not cylindrical. Indeed, when one removes a line from
and completes the resulting manifold at its positive end, the curve C becomes asymptotic to a curve \(C'\) with three ends each of multiplicity 1, rather than one end of multiplicity 3.
, and is tangent to the kernel of 
, where we quotient out by the finite dimensional group of biholomorphisms of the domain. In the situations considered here, this group is trivial whenever the total number of punctures or marked points on the domain is at least three.
, while a positive end is modelled on 
. Thus negative ends are concave (contracting as one moves towards 
in [
is an unbranched multiple cover of the cylinder 
over an asymptotic Reeb orbit 
and hence are related to numbers of integer points: see [
and completes the resulting manifold at its positive end, the curve C becomes asymptotic to a curve References
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Acknowledgements
This paper is an offshoot of my joint project with Dan Cristofaro-Gardiner and Richard Hind to calculate \(c_k(x)\) for all \(x>\tau ^4\). I wish to thank them for being such wonderful mathematicians to work with, Michael Hutchings for help with understanding obstruction bundle gluing, and finally the referee whose careful comments helped clarify the exposition. Also I wish to recognize Dr. Edge who taught me when I was an undergraduate at Edinburgh University in 1962–1967. He gave an inspiring course on projective geometry: I still remember him telling us that the secret of the 27 lines on the cubic surface lay in four dimensions—very mysterious to me at the time!
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Partially supported by NSF Grant DMS1308669.
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McDuff, D. A remark on the stabilized symplectic embedding problem for ellipsoids. European Journal of Mathematics 4, 356–371 (2018). https://doi.org/10.1007/s40879-017-0184-y
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DOI: https://doi.org/10.1007/s40879-017-0184-y