arXiv subjects

Joé Brendel

Publications and source records attributed to Joé Brendel.

12 recordsLinked to original sources

arXiv2026

The Lagrangian flux-monodromy morphism

We define a new morphism on the fundamental group of the space of Lagrangian submanifolds, which takes into account the Lagrangian flux and monodromy. We study the discreteness of its image and relate this discreteness to the ($C^\infty$) topology of the Hamiltonian orbit of the Lagrangian in question. We completely describe this new morphism for Lagrangian tori in $\mathbb{R}^4$ and $\mathbb{C}P^2$ and give explicit constructions. Similar constructions allow us to study the shape invariant of the Clifford torus in $\mathbb{C}P^n$. Finally, we upgrade our results on Lagrangian tori in $\mathbb{R}^4$ from the $C^\infty$ topology to the Hausdorff one.

math.SG↗

arXiv2026

The spread construction and non-uniqueness of visible Lagrangians

Every even symplectic Hirzebruch surface contains a Lagrangian torus and every odd symplectic Hirzebruch surface contains a Lagrangian Klein bottle as their respective real locus. From a toric point of view, this is a visible Lagrangian which surjects to the full moment polytope under the moment map. In this paper, we prove that every Hirzebruch surface contains both a Lagrangian torus and a Lagrangian Klein bottle with this property. An interesting consequence is that the topology of visible Lagrangian submanifolds is not determined by their image under the moment map. The proof is based on a new construction, which we call the spread of a family of Hamiltonian diffeomorphisms.

math.SG↗

arXiv2026

On Lagrangians of Oakley-Usher type

The first main theme of this paper is to study symplectic (un-)knottedness of Lagrangian submanifolds which are not tori. Our construction of such Lagrangians is inspired by the generalized Polterovich Lagrangians of Oakley--Usher and relies on an adaption of symplectic reduction methods going back to Chekanov--Schlenk and McDuff's probes to the case of non-abelian group actions. We compare Oakley--Usher Lagrangians to their (un-)knotted cousins to find that, depending on the situation, they are: (1) not diffeomorphic, (2) diffeomorphic, but not Lagrangian isotopic; (3) Lagrangian isotopic, but not Hamiltonian isotopic, (4) Hamiltonian isotopic. This relationship between them frequently changes when the ambient space is compactified. The second main theme is an in-depth study of monotone Lagrangian tori obtained from appyling our constructions to the three-dimensional quadric $Q^3$. Computing the versal deformation of the $Ψ$-invariant of Shelukhin--Tonkonog--Vianna, we find a torus which is knotted in a strong sense: it is not Hamiltonian isotopic to the Biran-lift of any Vianna torus in the two-dimensional quadric. Furthermore, we investigate enumerative properties of the Oakley--Usher torus in $Q^3$ and prove its non-displaceability, settling an open question asked by Oakley--Usher. We relate this to mirror symmetry by proving a split-generation result for the monotone Fukaya category of $Q^3$, which is inspired by, and can be compared to a similar result of Abouzaid--Diogo for cotangent bundles of the sphere.

math.SG↗

arXiv2026

A note on the cardinality of Lagrangian packings

Given a symplectic manifold, can one pack uncountably many Lagrangian submanifolds in a given Hamiltonian isotopy class of this symplectic manifold? We address $C^\infty$ and $C^0$ versions of this question.

math.SG↗

arXiv2025

Markov staircases

Rational homology ellipsoids are certain Liouville domains diffeomorphic to rational homology balls and having Lagrangian pin-wheels as their skeleta. From the point of view of almost toric fibrations, they are a natural generalisation of usual symplectic ellipsoids. We study symplectic embeddings of rational homology ellipsoids into the complex projective plane and we show that for each Markov triple, this problem gives rise to an infinite staircase. A key ingredient in the proof is the result that any two such embeddings are Hamiltonian isotopic. We also prove constraints on sizes for pairs of disjoint embeddings.

math.SG↗

arXiv2025

Lagrangian split tori in $S^2 \times S^2$ and billiards

In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in $S^2 \times S^2$, when the latter is equipped with a non-monotone split symplectic form. We show that this classification is equivalent to a problem of mathematical billiards in rectangles. We give many applications, among others: (1) answering a question on Lagrangian packing numbers raised by Polterovich--Shelukhin, (2) studying the topology of the space of Lagrangian tori, and (3) determining which split tori are images under symplectic ball embeddings of Chekanov or product tori in $\mathbb{R}^4$.

math.SG↗

arXiv2024

Local exotic tori

For a broad class of symplectic manifolds of dimension at least six, we find the following new phenomenon: there exist local exotic Lagrangian tori. More specifically, let $X$ be a geometrically bounded symplectic manifold of dimension at least six. We show that every open subset of $X$ contains infinitely many Lagrangian tori which are distinct up to symplectomorphisms of $X$ while being Lagrangian isotopic and having the same classical invariants. The proof relies on a locality property of the displacement energy germ, which allows us to compute it in a Darboux chart. Since these tori are not monotone, bubbling may occur and the count of Maslov index two $J$-holomorphic disks does not yield an invariant.

math.SG↗

arXiv2024

Semi-Local Exotic Lagrangian Tori in Dimension Four

We study exotic Lagrangian tori in dimension four. In certain Stein domains $B_{dpq}$ (which naturally appear in almost toric fibrations) we find $d+1$ families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of $B_{dpq}$ into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when $X$ is compact and (almost) toric and hence conclude that $X$ contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of $X$. In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces. Neither the embedded tori, nor the ambient space $X$ needs to be monotone for our methods to work.

math.SG↗

arXiv2023

Hamiltonian Classification of toric fibres and symmetric probes

In a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space. On the construction side of this question, we introduce a new method of constructing equivalences of toric fibres by using a symmetric version of McDuff's probes (see arXiv:0904.1686 and arXiv:1203.1074). On the other hand, we derive some obstructions to such equivalence by using Chekanov's classification of product tori together with a lifting trick from toric geometry. Furthermore, we conjecture that (iterated) symmetric probes yield all possible equivalences and prove this conjecture for $\mathbb{C}^n,\mathbb{C}P^2, \mathbb{C} \times S^2, \mathbb{C}^2 \times T^*S^1, T^*S^1 \times S^2$ and monotone $S^2 \times S^2$. This problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, i.e. determining which automorphisms of the homology of the toric fibre can be realized by a Hamiltonian diffeomorphism mapping the toric fibre in question to itself. For the above list of examples, we determine the Hamiltonian monodromy group for all toric fibres.

math.SG↗

arXiv2022

Pinwheels as Lagrangian barriers

The complex projective plane CP^2 contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of CP^2, meaning that pinwheels are Lagrangian barriers in the sense of Biran. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below 3.

math.SG↗

arXiv2020

Real Lagrangian Tori and Versal Deformations

Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called \emph{real}. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not real, extending Kim's result (arXiv:1909.09972) for $S^2 \times S^2$. Inside products of $S^2$, we show that all products of Chekanov tori are pairwise distinct and not real either. These results indicate that real tori are rare. Our methods are elementary in the sense that we do not use~$J$-holomorphic curves. Instead, we rely on symplectic reduction and the displacement energy of product tori in $\mathbb{R}^{2n}$.

math.SG↗

arXiv2019

On the topology of real Lagrangians in toric symplectic manifolds

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

math.SG↗