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Richard Hind

Publications and source records attributed to Richard Hind.

At least 19 recordsLinked to original sources
arXiv2026

Symplectic non-Kähler manifolds with and without the Hard Lefschetz Condition

In this paper we construct compact manifolds without Kähler structures that admit both a symplectic form satisfying the Hard Lefschetz Condition (HLC) and another symplectic form that does not. Our construction builds upon the orbifold introduced by Fernández and Muñoz and its symplectic resolution studied by Cavalcanti, Fernández, and Muñoz. By considering a one-parameter family of symplectic forms on the orbifold, we show that the corresponding resolved manifolds fail to satisfy the HLC for all parameters. However, after performing a suitable symplectic blowup along a union of tori, we obtain a family of symplectic manifolds for which the HLC holds for all non-zero parameters but fails at the central parameter. As a consequence, we exhibit a smooth manifold with no Kähler structure whose space of symplectic forms contains both HLC and non-HLC structures in the same connected component. This provides new examples of the subtle interplay between symplectic topology and the Hard Lefschetz property.

math.SG↗
arXiv2026

New constraints on Lagrangian embeddings and the shape invariant

For a large class of toric domains in $\mathbb{R}^4$ we determine which product Lagrangian tori can be mapped into the domain by a Hamiltonian diffeomorphism. In other words, we compute the Hamiltonian shape invariant of these toric domains, as defined by Hind and Zhang. The argument relies on new intersection results for product Lagrangian tori in symplectic polydisks. For Hamiltonian diffeomorphisms which map certain Lagrangian product tori back into the polydisk, we establish intersections between the images and a one-parameter family of product Lagrangian tori that includes (is based at) the original torus. For symplectic polydisks with area ratios less than two, we strengthen this to establish intersections between the Hamiltonian images and the original Lagrangian torus. As a soft complement to these intersection results we also present an embedding construction which demonstrates that this intersection rigidity vanishes when the one-parameter family of product Lagrangian tori is replaced by a natural packing by Lagrangian tori.

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arXiv2024

Symplectic field theory: an overview

We summarize some of the main ideas and results around symplectic field theory, from its early inception up to recent and ongoing developments.

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arXiv2024

On the Gromov width of complements of Lagrangian tori

An integral product Lagrangian torus in the standard symplectic $\mathbb{C}^2$ is defined to be a subset $\{ π|z_1|^2 = k, \, π|z_2|^2 =l \}$ with $k,l \in \mathbb{N}$. Let $\mathcal{L}$ be the union of all integral product Lagrangian tori. We compute the Gromov width of complements $B(R) \setminus \mathcal{L}$ for some small $R$, where $B(R)$ denotes the round ball of capacity $R$.

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arXiv2024

Quantitative Results on Symplectic Barriers

In this paper we present some quantitative results concerning symplectic barriers. In particular, we answer a question raised by Sackel, Song, Varolgunes, and Zhu regarding the symplectic size of the $2n$-dimensional Euclidean ball with a codimension-two linear subspace removed.

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arXiv2024

On the Existence of Symplectic Barriers

In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. More precisely, we prove that if a Euclidean ball is symplectically embedded in the Euclidean unit ball, then it must intersect a sufficiently fine grid of two-codimensional pairwise disjoint symplectic planes. Inspired by analogous terminology for Lagrangian submanifolds, we refer to these obstructions as symplectic barriers.

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arXiv2023

On the large-scale geometry of domains in an exact symplectic 4-manifold

We show that the space of open subsets of any complete and exact symplectic $4$-manifold has infinite dimension with respect to the symplectic Banach-Mazur distance; the quasi-flats we construct take values in the set of dynamically convex domains. In the case of $\mathbb{R}^4$, we therefore obtain the following contrast: the space of convex domains is quasi-isometric to a plane, while the space of dynamically convex ones has infinite dimension. In the case of $T^* S^2$, a variant of our construction resolves a conjecture of Stojisavljević and Zhang, asserting that the space of star-shaped domains in $T^* S^2$ has infinite dimension. Another corollary is that the space of contact forms giving the standard contact structure on $S^3$ has infinite dimension with respect to the contact Banach-Mazur distance.

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arXiv2023

Boundaries of open symplectic manifolds and the failure of packing stability

A finite volume symplectic manifold is said to have "packing stability" if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it has been conjectured that it always holds. We give counterexamples to this conjecture; in fact, we give examples that cannot be fully packed by any domain with smooth boundary nor by any convex domain. The examples are symplectomorphic to open and bounded domains in $\mathbb{R}^4$, with the diffeomorphism type of a disc. The obstruction to packing stability is closely tied to another old question, which asks to what extent an open symplectic manifold has a well-defined boundary; it follows from our results that many examples cannot be symplectomorphic to the interior of a compact symplectic manifold with smooth boundary. Our results can be quantified in terms of the volume decay near the boundary, and we produce, for example, smooth toric domains that are only symplectomorphic to the interior of a compact domain if the boundary of this domain has inner Minkowski dimension arbitrarily close to $4$. The growth rate of the subleading asymptotics of the ECH spectrum plays a key role in our arguments. We prove a very general "fractal Weyl law", relating this growth rate to the Minkowski dimension; this formula is potentially of independent interest.

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arXiv2023

On the agreement of symplectic capacities in high dimension

A theorem of Gutt-Hutchings-Ramos asserts that all normalized symplectic capacities give the same value for monotone four-dimensional toric domains. We generalize this theorem to arbitrary dimension. The new ingredient in our proof is the construction of symplectic embeddings of "$L$-shaped" domains in any dimension into corresponding infinite cylinders; this resolves a conjecture of Gutt-Pereira-Ramos in the affirmative.

math.SG↗
arXiv2023

Almost complex blow-ups and positive closed $(1,1)$-forms on $4$-dimensional almost complex manifolds

Let $(M,J)$ be a $2n$-dimensional almost complex manifold and let $x\in M$. We define the notion of almost complex blow-up of $(M,J)$ at $x$. We prove the existence of almost complex blow-ups at $x$ under suitable assumptions on the almost complex structure $J$ and we provide explicit examples of such a construction. We note that almost complex blow-ups are unique. When $(M,J)$ is a $4$-dimensional almost complex manifold, we give an obstruction on $J$ to the existence of almost complex blow-ups at a point and prove that the almost complex blow-up at a point of a compact almost Kähler manifold is almost Kähler.

math.DG↗
arXiv2022

Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids

The third named author has been developing a theory of "higher" symplectic capacities. These capacities are invariant under taking products, and so are well-suited for studying the stabilized embedding problem. The aim of this note is to apply this theory, assuming its expected properties, to solve the stabilized embedding problem for integral ellipsoids, when the eccentricity of the domain has the opposite parity of the eccentricity of the target and the target is not a ball. For the other parity, the embedding we construct is definitely not always optimal; also, in the ball case, our methods recover previous results of McDuff, and of the second named author and Kerman. There is a similar story, with no condition on the eccentricity of the target, when the target is a polydisc: a special case of this implies a conjecture of the first named author, Frenkel, and Schlenk concerning the rescaled polydisc limit function. Some related aspects of the stabilized embedding problem and some open questions are also discussed.

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arXiv2021

Families of almost complex structures and transverse $(p,p)$-forms

An {\em almost p-Kähler manifold} is a triple $(M,J,Ω)$, where $(M,J)$ is an almost complex manifold of real dimension $2n$ and $Ω$ is a closed real tranverse $(p,p)$-form on $(M,J)$, where $1\leq p\leq n$. When $J$ is integrable, almost $p$-Kähler manifolds are called $p$-{\em Kähler manifolds}. We produce families of almost $p$-Kähler structures $(J_t,Ω_t)$ on $\C^3$, $\C^4$, and on the real torus $\mathbb{T}^6$, arising as deformations of Kähler structures $(J_0,g_0,ω_0)$, such that the almost complex structures $J_t$ cannot be locally compatible with any symplectic form for $t\neq 0$. Furthermore, examples of special compact nilmanifolds with and without almost $p$-Kähler structures are presented.

math.DG↗
arXiv2021

Hamiltonian knottedness and lifting paths from the shape invariant

The Hamiltonian shape invariant of a domain $X \subset \mathbb R^4$, as a subset of $\mathbb R^2$, describes the product Lagrangian tori which may be embedded in $X$. We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when $X$ is a basic $4$-dimensional toric domain such as a ball $B^4(R)$, an ellipsoid $E(a,b)$ with $\frac{b}{a} \in {\mathbb N}_{\geq 2}$, or a polydisk $P(c,d)$. As applications, via the path lifting, we can detect knotted embeddings of product Lagrangian tori in many toric $X$. We also obtain novel obstructions to symplectic embeddings between domains that are more general than toric concave or toric convex.

math.SG↗
arXiv2021

The shape invariant of symplectic ellipsoids

The shape invariant of a symplectic manifold encodes the possible area classes of embedded Lagrangian tori. Potentially this is a powerful invariant, but for most manifolds the shape is unknown. We compute the shape for 4 dimensional symplectic ellipsoids where the ratio of the factors is an integer. The full shape invariant gives stronger embedding obstructions than results by considering only monotone tori.

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arXiv2020

On the Anti-Invariant Cohomology of Almost Complex Manifolds

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is infinite dimensional, and also $0$- or $1$-dimensional. In the compact case, we construct $6$-dimensional almost complex manifolds with arbitrary large anti-invariant cohomology and a $2$-parameter family of almost complex structures on the Kodaira-Thurston manifold whose anti-invariant cohomology group has maximum dimension.

math.DG↗
arXiv2017

On $L_2$-cohomology of almost Hermitian manifolds

Let $(X,J,ω,g)$ be a complete $n$-dimensional Kähler manifold. A Theorem by Gromov \cite{G} states that the if the Kähler form is $d$-bounded, then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless $k=\frac{n}{2}$. Starting with a contact manifold $(M,α)$ we show that the same conclusion does not hold in the category of almost Kähler manifolds. Let $(X,J,g)$ be a complete almost Hermitian manifold of dimension four. We prove that the reduced $L_2$ $2^{nd}$-cohomology group decomposes as direct sum of the closure of the invariant and anti-invariant $L_2$-cohomology. This generalizes a decomposition theorem by Drǎghici, Li and Zhang \cite{DLZ} for $4$-dimensional closed almost complex manifolds to the $L_2$-setting.

math.DG↗