Liouville Manifold Research Articles

On the embedding complexity of Liouville manifolds

On the embedding complexity of Liouville manifolds

On the Rouquier dimension of wrapped Fukaya categories and a conjecture of Orlov

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most$3$. As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric$3$-folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

Lagrangian cobordisms in Liouville manifolds

Floer theory for Lagrangian cobordisms was developed by Biran and Cornea in a series of papers [Lagrangian cobordism. I, J. Amer. Math. Soc. 26 (2013) 295–340; Lagrangian cobordism and Fukaya categories, Geom. Funct. Anal. 24 (2014) 1731–1830; Cone-decompositions of Lagrangian cobordisms in Lefschetz fibrations, Selecta Math. 23 (2017) 2635–2704] to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.

Relative growth rate and contact Banach–Mazur distance

In this paper, we define non-linear versions of Banach–Mazur distance in the contact geometry set-up, called contact Banach–Mazur distances and denoted by \(d_\mathrm{CBM}\). Explicitly, we consider the following two set-ups, either on a contact manifold \(W \times S^1\) where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of \(d_\mathrm{CBM}\) are different in these two cases. In the former case the inputs are (contact) star-shaped domains of \(W \times S^1\) which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach–Mazur distance \(d_\mathrm{CBM}\) defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg–Polterovich. The main results are the large-scale geometric properties in terms of \(d_\mathrm{CBM}\). In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou–Kashiwara–Schapira’s sheaf quantization.

Stabilized convex symplectic manifolds are Weinstein

We show that a stabilized convex symplectic (also called Liouville) manifold with the homotopy type of a half dimensional CW-complex is symplectomorphic to a flexible Weinstein manifold.

Super-rigidity of certain skeleta using relative symplectic cohomology

This paper uses relative symplectic cohomology, recently studied by Varolgunes, to understand rigidity phenomena for compact subsets of symplectic manifolds. As an application, we consider a symplectic crossings divisor in a Calabi–Yau symplectic manifold [Formula: see text] whose complement is a Liouville manifold. We show that, for a carefully chosen Liouville structure, the skeleton as a subset of [Formula: see text] exhibits strong rigidity properties akin to superheavy subsets of Entov–Polterovich. Along the way, we expand the toolkit of relative symplectic cohomology by introducing products and units. We also develop what we call the contact Fukaya trick, concerning the behavior of relative symplectic cohomology of subsets with contact type boundary under adding a Liouville collar.

The Structure of the Conjugate Locus of a General Point on Ellipsoids and Certain Liouville Manifolds

It is well known since Jacobi that the geodesic flow of the ellipsoid is "completely integrable", which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior of the geodesics becomes easy to see. In fact, it happened quite recently that a proof for the statement "The conjugate locus of a general point in two-dimensional ellipsoid has just four cusps" in Jacobi's Vorlesungen \"uber dynamik appeared in the literature. In this paper, we consider Liouville manifolds, a certain class of Riemannian manifolds which contains ellipsoids. We solve the geodesic equations; investigate the behavior of the Jacobi fields, especially the positions of the zeros; and clarify the structure of the conjugate locus of a general point. In particular, we show that the singularities arising in the conjugate loci are only cuspidal edges and $D_4^+$ Lagrangian singularities, which would be the higher dimensional counterpart of Jacobi's statement.

Covariantly functorial wrapped Floer theory on Liouville sectors

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid's generation criterion follows.

Koszul duality via suspending Lefschetz fibrations

Let $M$ be a Liouville 6-manifold which is the smooth fiber of a Lefschetz fibration on $\mathbb{C}^4$ constructed by suspending a Lefschetz fibration on $\mathbb{C}^3$. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category $\mathcal{F}(M)$ and the wrapped Fukaya category $\mathcal{W}(M)$ are related through $A_\infty$-Koszul duality, by identifying them with cyclic and Calabi-Yau completions of the same quiver algebra. This implies the split-generation of the compact Fukaya category $\mathcal{F}(M)$ by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi-dilations in the sense of Seidel-Solomon are obtained.

Symplectic homology of complements of smooth divisors

If ( X , ω ) is a closed symplectic manifold, and Σ is a smooth symplectic submanifold Poincaré dual to a positive multiple of ω, then X ∖ Σ can be completed to a Liouville manifold ( W , d λ ) . Under monotonicity assumptions on X and on Σ, we construct a chain complex whose homology computes the symplectic homology of W. We show that the differential is given in terms of Morse contributions, Gromov–Witten invariants of X relative to Σ and Gromov–Witten invariants of Σ. We use a Morse–Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to Σ.

Orderability, contact non-squeezing, and Rabinowitz Floer homology

We study Liouville fillable contact manifolds $(\Sigma,\xi)$ with non-zero Rabinowitz Floer homology and assign spectral numbers to paths of contactomorphisms. As a consequence we prove that $\widetilde{\mathrm{Cont}_0}(\Sigma,\xi)$ is orderable in the sense of Eliashberg and Polterovich. This provides a new class of orderable contact manifolds. If the contact manifold is in addition periodic or a prequantization space $M \times S^1$ for $M$ a Liouville manifold, then we construct a contact capacity. This can be used to prove a general non-squeezing result, which amongst other examples in particular recovers the beautiful non-squeezing results from [EKP06].

The stable Morse number as a lower bound for the number of Reeb chords

Assume that we are given a closed chord-generic Legendrian sub-manifold Lambda subset of P x R of the contactisation of a Liouville manifold, where Lambda moreover admits an exact Lagrangian fillin ...

Contact non-squeezing at large scale in ℝ2n × S1

We define a [Formula: see text]-equivariant version of the cylindrical contact homology used by Eliashberg–Kim–Polterovich [11] to prove contact non-squeezing for prequantized integer-capacity balls [Formula: see text], [Formula: see text] and we use it to extend their result to all [Formula: see text]. Specifically, we prove if [Formula: see text] there is no [Formula: see text], the group of compactly supported contactomorphisms of [Formula: see text] which squeezes [Formula: see text] into itself, i.e. maps the closure of [Formula: see text] into [Formula: see text]. A sheaf theoretic proof of non-existence of corresponding [Formula: see text], the identity component of [Formula: see text], is due to Chiu [7] it is not known if this is strictly weaker. Our construction has the advantage of retaining the contact homological viewpoint of Eliashberg–Kim–Polterovich and its potential for application in prequantizations of other Liouville manifolds. It makes use of the [Formula: see text]-action generated by a vertical [Formula: see text]-shift but can also be related, for prequantized balls, to the [Formula: see text]-equivariant contact homology developed by Milin [16] in her proof of orderability of lens spaces.

Floer theory for Lagrangian cobordisms

In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several long exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms.

Thread construction revisited

Staude's thread construction of ellipsoid is revisited from a new view-point concerning the length of geodesic segments. Thanks to the general nature of this view-point, one obtains similar thread construction on other stages, i.e., on “Liouville manifolds”.

A Liouville property for gradient graphs and a Bernstein problem for Hamiltonian stationary equations

Using a rotation of Yuan, we observe that the gradient graph of any semi-convex function is a Liouville manifold, that is, does not admit non-constant bounded harmonic functions. As a corollary, we find that any solution of the fourth order Hamiltonian stationary equation satisfying $$\theta \geq \left( n - 2\right) \frac{\pi}{2} + \delta$$ for some \({\delta > 0}\) must be a quadratic.

Exotic iterated Dehn twists

Consider cotangent bundles of exotic spheres, with their canonical symplectic structure. They admit automorphisms which preserve the part at infinity of one fibre, and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all automorphisms with the same behaviour).

Bounding Lagrangian widths via geodesic paths

AbstractThe width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg–Murphy. In this paper we introduce a new method for bounding the width of a Lagrangian$Q$by considering the Lagrangian Floer cohomology of an auxiliary Lagrangian$L$with respect to a Hamiltonian whose chords correspond to geodesic paths in$Q$. This is formalized as a wrapped version of the Floer–Hofer–Wysocki capacity and we establish an associated energy–capacity inequality with the help of a closed–open map. For any orientable Lagrangian$Q$admitting a metric of non-positive sectional curvature in a Liouville manifold, we show the width of$Q$is bounded above by four times its displacement energy.

C-projective equivalence and integrability of the geodesic flow

We show that there is a deep relation between C -projective equivalence and Kähler–Liouville manifolds; the latter is a typical class of Kähler manifolds whose geodesic flows are integrable in Liouville’s sense.

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.