Quantitative embedded contact homology

By Michael Hutchings. 2010

165 citations recorded by the source corpus.

Define a "Liouville domain" to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset thereof) a sequence of real numbers, which we call "ECH capacities". The ECH capacities of a Liouville domain are defined in terms of the "ECH spectrum" of its boundary, which measures the amount of symplectic action needed to represent certain classes in embedded contact homology. Using cobordism maps on embedded contact homology (defined in joint work with Taubes), we show that the ECH capacities are monotone with respect to symplectic embeddings. We compute the ECH capacities of ellipsoids, polydisks, certain subsets of the cotangent bundle of T2, and disjoint unions of examples for which the ECH capacities are known. The resulting symplectic embedding obstructions are sharp in some interesting cases, for example for the problem of embedding an ellipsoid into a ball (as shown by work of McDuff-Schlenk) or embedding a disjoint union of balls into a ball. We also state and present evidence for a conjecture under which the asymptotics of the ECH capacities of a Liouville domain recover its symplectic volume.

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

View Paper: https://www.semanticscholar.org/paper/1b4dc37dafc4c8b95e49130dd3f71b3cae0c928c

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Citations in the corpus

The Gromov width of 4-dimensional tori (2011, background): “oblem of embedding a polydisc cannot be reduced to the problem of embedding a collection of balls. Ekeland–Hofer capacities, [14], and the new 4-dimensional invariants from embedded contact homology, [19], provide obstructions for symplectic embeddings of polydiscs into certain manifolds. A few symp”
The asymptotics of ECH capacities (2012, background): “In [4], a sequence of real numbers 0 = c0(X,ω) < c1(X,ω) ≤ c2(X,ω) ≤ · · · ≤ ∞ ∗The first author was partially supported by NSF grant DMS-0838703.”
The embedding capacity of 4-dimensional symplectic ellipsoids (2009, methodology): “Using work by Taubes, Hutchings shows in [10] that the sequence N(a, b) is a monotone invariant of E(a, b):”
Symplectic embeddings from concave toric domains into convex ones (2014, methodology): “The fact that a symplectic embedding int(XΩ1)→ int(XΩ2) implies that ck(int(XΩ1)) ≤ ck(int(XΩ2)) for all k follows from the Monotonicity property of ECH capacities shown in [6].”
The Hofer conjecture on embedding symplectic ellipsoids (2010, background): “ss than or equal to N(c,d) (written N(a,b) 4 N(c,d)) if, for all k ≥ 0, the kth entry of N(a,b) is at most equal to the kth entry in N(c,d). Using embedded contact homology (ECH), Hutchings showed in [1] that if there is a symplectic embedding intE(a,b) s →intE(c,d) then N(a,b) 4 N(c,d). Hofer’s con”
Dense existence of periodic Reeb orbits and ECH spectral invariants (2015, unknown): “For any σ ∈ ECH∗(Y, ξ,Γ) \ {0}, let us recall from [5] the definition of cσ(Y, λ) ∈ R≥0.”
Contact geometry and isosystolic inequalities (2011, background): “5 in [16]) implies that the systolic constant of the standard contact structure on the three-torus is greater than 1/2.”
Proof of the Arnold chord conjecture in three dimensions I (2011, background): “This result also has additional applications, for example it gives rise to new obstructions to symplectic embeddings in four dimensions; see Hutchings [8].”
Symplectic capacities from positive S1–equivariant symplectic homology (2017, methodology): “tcEH i pXq c j pX 1qu; (1.5) where iand jare nonnegative integers and we interpret cEH 0 0. More recently, embedded contact homology was used to dene the ECH capacities of symplectic four-manifolds [Hut11a]. ECH capacities can be computed in many examples, such as four-dimensional \concave toric dom”
Proof of the Arnold chord conjecture in three dimensions, II (2010, methodology): “ to prove the Arnold chord conjecture in three dimensions. This result also has additional applications, for example it gives rise to new obstructions to symplectic embeddings in four dimensions, see [7]. Along the way to proving Theorem 1.9, we will also prove that ﬁltered ECH does not depend on th”
Lecture Notes on Embedded Contact Homology (2013, background): “10) (19)The definition of XΩ in [24] is different, but symplectomorphic to the one given here.”
Punctured holomorphic curves and Lagrangian embeddings (2014, background): “Most known symplectic capacities (exceptions being the higher Ekeland–Hofer capacities [24] for subsets of C and the ECH capacities [39] in dimension 4) take the same value on the unit ball and on the cylinder.”
Beyond ECH capacities (2014, methodology): “In [8], embedded contact homology (ECH) was used to define, for any symplectic four-manifold with boundary X , a sequence of real numbers 0 = c0(X) ≤ c1(X) ≤ c2(X) ≤ · · · ≤ ∞ called ECH capacities , such that if X symplectically embeds into X ′ then ck(X) ≤ ck(X ) for all k.”
From symplectic measurements to the Mahler conjecture (2013, background): “Since Gromov’s work, several other symplectic capacities were constructed, such as the Hofer–Zehnder capacity [15], Ekeland–Hofer capacities [6], the displacement energy [14], spectral capacities [7, 23, 28], and Hutchings’ embedded contact homology capacities [16].”
Symplectic embedding problems, old and new (2017, background): “Among the unforgivable omissions are the breakthrough in the problem of ball packing stability in higher dimensions by Buse and Hind [17,18] and Hutchings’s ECH capacities [62], which form a whole sequence of symplectic embedding invariants of 4-dimensional domains, that provide a complete set of ob”
Symplectic embeddings into four‐dimensional concave toric domains (2013, background): “1 requires some knowledge of the definition of ECH capacities, which we now briefly review; for full details, see [5] or [7].”