arXiv subjects

Oliver Edtmair

Publications and source records attributed to Oliver Edtmair.

11 recordsLinked to original sources

arXiv2025

Smooth perfectness of Hamiltonian diffeomorphism groups

A fundamental result of Banyaga states that the Hamiltonian diffeomorphism group of a closed symplectic manifold is perfect. We refine this result by proving that, locally in the $C^\infty$ topology, the number of commutators needed to express a Hamiltonian diffeomorphism is bounded, and the commutators can be chosen to depend smoothly on the diffeomorphism. As a corollary, we show that any homogeneous quasimorphism on the Hamiltonian diffeomorphism group is continuous in the $C^\infty$ topology. We establish an analogous smooth perfectness result for compactly supported Hamiltonian diffeomorphisms of open symplectic manifolds, where Banyaga proved that the kernel of the Calabi homomorphism is perfect. For symplectic manifolds with boundary, we define a natural group of Hamiltonian diffeomorphisms that need not restrict to the identity on the boundary. We extend the Calabi homomorphism to this setting and prove a corresponding refined perfectness result. These results are motivated by our work on symplectic packing stability for symplectic manifolds with boundary, where they play a key role in our symplectic embedding constructions.

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arXiv2025

Packing stability and the subleading asymptotics of symplectic Weyl laws

We prove that symplectic ball packing stability holds for every compact, connected symplectic $4$-manifold with smooth boundary. This follows from a stronger result: the full volume of any such manifold can be filled by a single symplectic ellipsoid. As an application, we obtain estimates - with sharp exponents - for the error terms in the symplectic Weyl laws for embedded contact homology capacities, periodic Floer homology spectral invariants, and link spectral invariants. We also construct an example of a star-shaped domain in $\mathbb{R}^4$, arbitrarily $C^1$ close to the unit ball and with boundary of regularity just below $C^2$ and smooth away from a single point, for which packing stability fails. Our proofs reveal a close connection between symplectic packing stability in the presence of smooth boundary and the algebraic structure of Hamiltonian diffeomorphism groups, particularly Banyaga's simplicity results.

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arXiv2025

A universal extension of helicity to topological flows

Helicity is a fundamental conserved quantity in physical systems governed by vector fields whose evolution is described by volume-preserving transformations on a three-manifold. Notable examples include inviscid, incompressible fluid flows, modeled by the three-dimensional Euler equations, and conducting plasmas, described by the magnetohydrodynamics (MHD) equations. A key property of helicity is its invariance under volume-preserving diffeomorphisms. In an influential article from 1973, Arnold, having provided an ergodic interpretation of helicity as the "asymptotic Hopf invariant", posed the question of whether this invariance persists under volume-preserving homeomorphisms. More generally, he asked whether helicity can be extended to topological volume-preserving flows. We answer both questions affirmatively for flows without rest points. Our approach reformulates Arnold's question in the framework of what we call $C^0$ Hamiltonian structures. This perspective enables us to leverage recent developments in $C^0$ symplectic geometry, particularly results concerning the algebraic structure of the group of area-preserving homeomorphisms.

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arXiv2024

On closed characteristics of minimal action on a convex three-sphere

We prove that every closed characteristic of minimal action on the boundary of a uniformly convex domain in $\R^4$ bounds a disk-like global surface of section. A corollary is that the cylindrical symplectic capacity of a convex body in $\R^4$ coincides with the minimal action of a closed generalized characteristic on its boundary.

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arXiv2023

Symplectic capacities of domains close to the ball and Banach-Mazur geodesics in the space of contact forms

We prove that all normalized symplectic capacities coincide on smooth domains in $\mathbb C^n$ which are $C^2$-close to the Euclidean ball, whereas this fails for some smooth domains which are just $C^1$-close to the ball. We also prove that all symplectic capacities whose value on ellipsoids agrees with that of the $n$-th Ekeland-Hofer capacity coincide in a $C^2$-neighborhood of the Euclidean ball of $\mathbb C^n$. These results are deduced from a general theorem about contact forms which are $C^2$-close to Zoll ones, saying that these contact forms can be pulled back to suitable "quasi-invariant" contact forms. We relate all this to the question of the existence of minimizing geodesics in the space of contact forms equipped with a Banach-Mazur pseudo-metric. Using some new spectral invariants for contact forms, we prove the existence of minimizing geodesics from a Zoll contact form to any contact form which is $C^2$-close to it. This paper also contains an appendix in which we review the construction of exotic ellipsoids by the Anosov-Katok conjugation method, as these are related to the above mentioned pseudo-metric.

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arXiv2023

Legendrian embedded contact homology

We give a construction of embedded contact homology (ECH) for a contact $3$-manifold $Y$ with convex sutured boundary and a pair of Legendrians $\Lambda_+$ and $\Lambda_-$ contained in $\partial Y$ satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of $Y$ and Reeb chords of $\Lambda_+$ to $\Lambda_-$. The main ingredients include: a general Legendrian adjunction formula for curves in $\mathbb{R} \times Y$ with boundary on $\mathbb{R} \times \Lambda$; a relative writhe bound for curves in contact $3$-manifolds asymptotic to Reeb chords; and a Legendrian ECH index with an accompanying ECH index inequality. The (action filtered) Legendrian ECH of any pair $(Y,\Lambda)$ of a closed contact $3$-manifold $Y$ and a Legendrian link $\Lambda$ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in Colin-Ghiggini-Honda's proof of the equivalence of Heegaard-Floer homology and ECH. The independence of our construction of choices of almost complex structure and contact form should require a new flavor of monopole Floer homology. It is beyond the scope of this paper.

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arXiv2022

An elementary alternative to PFH spectral invariants

Inspired by Hutchings' elementary alternative to ECH capacities, we introduce an elementary alternative to spectral invariants defined via periodic Floer homology (PFH). We use these spectral invariants to provide more elementary proofs of a number of results which have recently been obtained using PFH spectral invariants. Among these results are quantitative closing lemmas for area-preserving surface diffeomorphisms and the simplicity conjecture.

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arXiv2022

Disk-like surfaces of section and symplectic capacities

We prove that the cylindrical capacity of a dynamically convex domain in $\mathbb{R}^4$ agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in $\mathbb{R}^4$ which are sufficiently $C^3$ close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salom\~{a}o establishing a systolic inequality for such domains.

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arXiv2022

The Ruelle Invariant And Convexity In Higher Dimensions

We construct the Ruelle invariant of a volume preserving flow and a symplectic cocycle in any dimension and prove several properties. In the special case of the linearized Reeb flow on the boundary of a convex domain $X$ in $\mathbb{R}^{2n}$, we prove that the Ruelle invariant $\text{Ru}(X)$, the period of the systole $c(X)$ and the volume $\text{vol}{X}$ satisfy \[\text{Ru}(X) \cdot c(X) \le C(n) \cdot \text{vol}{X}\] Here $C(n) > 0$ is an explicit constant dependent on $n$. As an application, we construct dynamically convex contact forms on $S^{2n-1}$ that are not convex, disproving the equivalence of convexity and dynamical convexity in every dimension.

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arXiv2022

3d Convex Contact Forms And The Ruelle Invariant

Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomão. These are the first examples of dynamically convex contact $3$-spheres that are not strictly contactomorphic to a convex boundary $Y$.

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arXiv2021

PFH spectral invariants and $C^\infty$ closing lemmas

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving surface diffeomorphisms. In particular, we show that for a $C^\infty$-generic area-preserving diffeomorphism of the torus, the set of periodic points is dense. Our closing lemmas are quantitative, asserting roughly speaking that for a given Hamiltonian isotopy, within time $\delta$ a periodic orbit must appear of period $O(\delta^{-1})$. We also prove a "Weyl law" describing the asymptotic behavior of PFH spectral invariants.

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