arXiv subjects

Francisco Presas

Publications and source records attributed to Francisco Presas.

At least 19 recordsLinked to original sources

arXiv2026

Spaces of tight contact structures in dimension 3

We study spaces of tight contact structures and contactomorphism groups in dimension $3$. We introduce a microfibration method for studying spaces of convex surfaces in contact $3$-manifolds and apply it to describe spaces of convex disks and spheres in tight contact $3$-manifolds. With these ingredients in hand, we prove a parametric gluing theorem for contact handle attachments: the weak homotopy type of the space of tight contact structures is invariant under contact $0$-, $1$-, and $3$-handle attachments. Based on these new techniques, we present a substantial number of independent applications. First, we obtain explicit descriptions of the weak homotopy type of several spaces of tight contact structures. We determine the weak homotopy type of the contactomorphism group of every standard contact handlebody, of the standard $\mathbb{S}^1\times\mathbb{S}^2$, and of every Legendrian circle bundle with non-empty boundary, partially resolving a conjecture of Giroux. Additionally, we compute the weak homotopy type of an infinite family of spaces of Legendrian and transverse unknots, proving Legendrian and transverse analogues of Hatcher's theorem for smooth unknots in the $3$-sphere.

math.SG↗

arXiv2026

Symplectic foliated fillings of sphere cotangent bundles

We classify symplectically foliated fillings of certain foliated manifolds with a contact structure on the leaves. We show that for the foliated sphere cotangent bundle of the Reeb foliation on the three-sphere, the corresponding foliated disk cotangent bundle is the unique strong symplectic foliated filling up to blowups and symplectic deformation equivalence. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with an almost horizontal foliation. In this case, the foliated filling of the foliated sphere cotangent bundle is not unique. We show that any such filling is a foliated Lefschetz fibration, and is determined up to symplectic deformation equivalence, by combinatorial invariants arising from the singular locus of the Lefschetz fibration.

math.SG↗

arXiv2025

Convex disks with Legendrian boundary in overtwisted contact 3-manifolds

We classify convex disks with a fixed characteristic foliation and Legendrian boundary, up to contact isotopy relative to the boundary, in every closed overtwisted contact 3-manifold. This classification covers cases where the neighborhood of such a disk is tight or where the boundary violates the Bennequin-Eliashberg inequality. We show that this classification coincides with the formal one, establishing an h-principle for these disks. As a corollary, we deduce that the space of Legendrian unknots that lie in some Darboux ball in a closed overtwisted contact 3-manifold satisfies the h-principle at the level of fundamental groups. Finally, we determine the contact mapping class group of the complement of each Legendrian unknot with non-positive tb invariant in an overtwisted 3-sphere.

math.GT↗

arXiv2024

Parametric satellites and connected-sums in the space of Legendrian embeddings

This article introduces two new constructions at the higher homotopy level in the space of Legendrian embeddings in $(\mathbb{R}^3, ξ_{\operatorname{std}})$. We first introduce the parametric Legendrian satellite construction, showing that the satellite operation works for parametric families of Legendrian embeddings. This yields new invariants at the higher-order homotopy level. We then introduce the parametric connected-sum construction. This operation takes as inputs two $n$-spheres based at Legendrian embeddings $K_1$ and $K_2$, respectively, and produces a new $n$-sphere based at $K_1\# K_2$. As a main application we construct new infinite families of loops of Legendrian embeddings with non-trivial LCH monodromy invariant.

math.SG↗

arXiv2024

An $h$-principle for embeddings transverse to a contact structure

Given a class of embeddings into a contact or a symplectic manifold, we give a sufficient condition, that we call isocontact or isosymplectic realization, for this class to satisfy a general $h$-principle. The flexibility follows from the $h$-principles for isocontact and isosymplectic embeddings, it provides a framework for classical results, and we give two new applications. Our main result is that embeddings transverse to a contact structure satisfy a full $h$-principle in two cases: if the complement of the embedding is overtwisted, or when the intersection of the image of the formal derivative with the contact structure is strictly contained in a proper symplectic subbundle. We illustrate the general framework on symplectic manifolds by studying the universality of Hamiltonian dynamics on regular level sets via a class of embeddings.

math.SG↗

arXiv2023

Universality of Euler flows and flexibility of Reeb embeddings

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows,i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature.

math.DS↗

arXiv2021

Symplectic embeddings of 4-manifolds via Lefschetz fibrations

In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, ω_{pr}),$ where $ω_{pr}$ is the product symplectic form on $\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$--manifold in which all finite type Weinstein $4$--manifolds admit iso-symplectic embeddings.

math.GT↗

arXiv2021

Loose Engel structures

This article introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete h-principle when some auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

math.SG↗

arXiv2020

Constructing Turing complete Euler flows in dimension $3$

Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable of performing computations. More recently, Tao launched a programme based on the Turing completeness of the Euler equations to address the blow up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem [7] to quantum field theories [11]. To the best of our knowledge, the existence of undecidable particle paths of 3D fluid flows has remained an elusive open problem since Moore's works in the early 1990's. In this article we construct a Turing complete stationary Euler flow on a Riemannian $S^3$ and speculate on its implications concerning Tao's approach to the blow up problem in the Navier-Stokes equations.

math.DS↗

arXiv2020

Fundamental groups of formal Legendrian and horizontal embedding spaces

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of non trivial loops of Legendrian embeddings are already non trivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that this computes the fundamental group of the space of horizontal embeddings as well.

math.SG↗

arXiv2019

The Legendrian Whitney trick

In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.

math.SG↗

arXiv2018

Geometric criteria for overtwistedness

We establish geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate the different overtwisted disks in each dimension and show that a manifold is overtwisted if the Legendrian unknot is loose. Then we characterize overtwistedness in terms of open book decompositions and provide several applications.

math.SG↗

arXiv2018

Loops of Legendrians in contact 3-manifolds

We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.

math.SG↗

arXiv2018

The foliated Lefschetz hyperplane theorem

A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $ω$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of codimension--$2$ can be found for $(M, \mathcal{F}, ω)$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs $(F, F \cap W_k)$, with $F$ any leaf of $\mathcal{F}$. This is applied to draw important consequences on the transverse geometry of such foliations.

math.DG↗

arXiv2018

Notes On Open Book Decompositions For Engel Structures

We relate open book decompositions of a 4-manifold M with its Engel structures. Our main result is, given an open book decomposition of M whose binding is a collection of 2-tori and whose monodromy preserves a framing of a page, the construction of an En-gel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding. In particular the pages are contact man-ifolds and the monodromy is a contactomorphism. As a consequence, on a parallelizable closed 4-manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that amongst the supported Engel structures we construct, there is a class of loose Engel structures.

math.SG↗

arXiv2018

Tight neighborhoods of contact submanifolds

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.

math.SG↗

arXiv2017

Geometric quantization of semitoric systems and almost toric manifolds

Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed counting integral points inside the associated Delzant polytope. In this article we extend Kostant's geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one associated to an integrable system with focus-focus type singularities in dimension four. As application we determine models for the geometric quantization of K3 surfaces, a spin-spin system, the spherical pendulum, and a spin-oscillator system under this scheme.

math.SG↗

arXiv2016

A simple construction of positive loops of Legendrians

We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu's result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.

math.SG↗