Papers by Francisco Presas
arXiv (Cornell University), Nov 5, 2019
The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by... more The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [35, 36] launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. In this article we prove that the Euler equations exhibit universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are stationary of Beltrami type, so they exist for all time. Using this result, we establish the Turing completeness of the Euler flows, i.e. that 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 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.
Springer Proceedings in Physics, 2019
We study homotopically non-trivial spheres of Legendrians in the standard contact R 3 and S 3. We... more We study homotopically non-trivial spheres of Legendrians in the standard contact R 3 and S 3. We prove that there is a homotopy injection of the contactomorphism group of S 3 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.
Journal of Symplectic Geometry, 2020
Kostant gave a model for the geometric quantization via the cohomology associated to the sheaf of... more Kostant gave a model for the geometric quantization 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 by 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 induced by an integrable system with focus-focus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme.
arXiv: Symplectic Geometry, 2018
We classify symplectically foliated fillings of certain contact foliated manifolds. We show that ... more We classify symplectically foliated fillings of certain contact foliated manifolds. We show that up to symplectic deformation, the unique minimal symplectically foliated filling of the foliated sphere cotangent bundle of the Reeb foliation in the 3-sphere is the associated disk cotangent bundle. En route to the proof, we study another foliated manifold, namely the product of a circle and an annulus with almost horizontal foliation. In this case, the foliated unit cotangent bundle does not have a unique minimal symplectic filling. We classify the foliated fillings of this manifold up to symplectic deformation equivalence using combinatorial invariants of the filling.
Proceedings of the National Academy of Sciences, 2021
Significance What kind of physics might be noncomputational? In this article, we address the appe... more Significance What kind of physics might be noncomputational? In this article, we address the appearance of undecidable phenomena in fluid dynamics, proving the existence of Turing complete fluid flows on a three-dimensional geometric domain. Our strategy fuses the computational power of symbolic dynamics with geometric techniques from contact topology and its connection with hydrodynamics unveiled by Sullivan, Etnyre, and Ghrist more than two decades ago. The type of flows that we consider are stationary solutions to the incompressible Euler equations. Taking these steady flows as initial data, we obtain global solutions to the Navier–Stokes equations that can simulate just a (fixed) finite number of steps of any Turing machine. This raises the challenging question of the Turing-completeness of Navier–Stokes.
Journal of Symplectic Geometry, 2020
We prove that any small enough neighborhood of a closed contact submanifold is always tight under... more 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.
Algebraic & Geometric Topology, 2020
We compute the fundamental group of each connected component of the space of formal Legendrian em... more We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R 3. 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 R 4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at π1-level.
Revista Matemática Iberoamericana, 2018
We prove the existence of minimal symplectomorphisms and strictly ergodic contactomorphisms on ma... more We prove the existence of minimal symplectomorphisms and strictly ergodic contactomorphisms on manifolds which admit a locally free S 1-action by symplectomorphisms and contactomorphisms, respectively. The proof adapts the conjugation method, introduced by Anosov and Katok, to the contact and symplectic setting.
Nagoya Mathematical Journal, 2017
A foliation $(M,{\mathcal{F}})$ is said to be $2$ -calibrated if it admits a closed 2-form $\unic... more A foliation $(M,{\mathcal{F}})$ is said to be $2$ -calibrated if it admits a closed 2-form $\unicode[STIX]{x1D714}$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_{k}$ of $2$ -calibrated submanifolds of codimension- $2$ can be found for $(M,{\mathcal{F}},\unicode[STIX]{x1D714})$ . 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.
Inventiones mathematicae, 2017
International Mathematics Research Notices, 2017
A foliation is said to admit a foliated contact structure if there is a codimension 1 distributio... more A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the presence of an overtwisted leaf. The result is shown to be sharp.
Journal of Symplectic Geometry, 2016
Given an overtwisted contact manifold, we prove that there can be no positive loops of contactomo... more Given an overtwisted contact manifold, we prove that there can be no positive loops of contactomorphisms that are generated by a C 0-small Hamiltonian function.
Commentarii Mathematici Helvetici, 2016
In this article we address the existence of positive loops of contactomorphisms in overtwisted co... more In this article we address the existence of positive loops of contactomorphisms in overtwisted contact 3-folds. We present a construction of such positive loops in the contact fibered connected sum of certain contact 3-folds along transverse knots. In particular, we obtain positive loops of contactomorphisms in a class of overtwisted contact structures.
Journal of Geometry and Physics, 2017
We define Maslov-type indices associated to contact and symplectic transformation groups. There a... more We define Maslov-type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of Z. These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these indices and their applications.
Asian Journal of Mathematics, 2002
We define the concept of Lefschetz contact pencil and we show the existence of such structures on... more We define the concept of Lefschetz contact pencil and we show the existence of such structures on any contact manifold. The main idea of the proof is a generalization of the Donaldson arguments used in the symplectic case. We will analyze some of the applications of such existence theorem for the topology of approximately holomorphic contact submanifolds. In particular, we study the topological relationship between the contact submanifods constructed in [IMP99].
Revista Espanola De Cirugia Oral Y Maxilofacial Publicacion Oficial De La Sociedad Espanola De Cirugia Oral Y Maxilofacial, 2000
La hiperplasia condilar es un transtorno adquirido del desarrollo mandibular, de etiologia descon... more La hiperplasia condilar es un transtorno adquirido del desarrollo mandibular, de etiologia desconocida, poco frecuente y con una clinica caracteristica de deformidad en el tercio inferior de la cara y alteraciones oclusales y de articulacion temporomandibular. En este trabajo revisamos la bibliografia y nuestra experiencia sobre la hiperplasia condilar activa. Presentamos una serie de siete casos tratados en los ultimos anos, con un seguimiento postoperatorio que oscila entre los 54 y los 27 meses. Del total, cinco casos (71%) presentaban sintomatologia en la articulacion temporomandibular. A todos ellos se les aplico un protocolo diagnostico que incluye exploracion clinica rigurosa, estudio radiologico mediante ortopantomografia, telerradiografia, telerradiografia lateral de craneo con cefalometria y el diagnostico de confirmacion de las formas activas mediante gammagrafia con tecnecio 99. En todos los pacientes se realizo condilectomia alta y reposicionamiento del disco desinsectado, con fijacion mediante Mitek mini archor, y tratamiento simultaneo ortodoncio/quirurgico. Los resultados obtenidos han sido buenos, obteniendo una adecuada simetria facial al corregir la deformidad facial, una correcta oclusion y desaparicion de la sintomatologia articular previa en todos los casos.
Codimension one foliated manifolds $(M,\mathcal{F})$ admitting a closed 2-form $\omega$ making ea... more Codimension one foliated manifolds $(M,\mathcal{F})$ admitting a closed 2-form $\omega$ making each leaf symplectic are a natural generalization of 3-dimensional taut foliations. Remarkably, on such closed foliated manifolds $(M,\mathcal{F})$ there exists a class of 3-dimensional transverse closed submanifolds $W$ on which $\mathcal{F}$ induces a taut foliation $\mathcal{F}_W$. Our main result says that the foliated submanifold $(W,\mathcal{F}_W)$ has the same transverse geometry as $(M,\mathcal{F})$. More precisely, the inclusion induces an essential equivalence between the corresponding holonomy groupoids. The proof of our main result relies on a leafwise Lefschetz hyperplane theorem, which is of independent interest.
Journal of Symplectic Geometry, 2014
We prove the non-triviality of the Reeb flow for the standard contact spheres S 2n+1 , n = 3, ins... more We prove the non-triviality of the Reeb flow for the standard contact spheres S 2n+1 , n = 3, inside the fundamental group of their contactomorphism group. The argument uses the existence of homotopically non-trivial 2-spheres in the space of contact structures of a 3-Sasakian manifold. Let (M, ξ) be a closed contact manifold. Consider the space C(M, ξ) of contact structures isotopic to ξ. This space has been studied in special cases. See [El] for the 3-sphere and [Bo], [Ge] for torus bundles. In the present note we prove the non-triviality of its second homotopy group for 3-Sasakian manifolds, see [BG]. Theorem 1. Let (M, ξ) be a 3-Sasakian manifold, then rk(π 2 (C(M, ξ))) ≥ 1. Let (S 4n+3 , ξ 0 = ker α 0) be the standard contact sphere with the standard contact form. The nontrivial spheres in C(S 4n+3 , ξ 0) allow us to answer a question posed in [Gi]: Remarque 2.10: On peut se demander s'il n'y a pas, dans Cont(S 2n+1 , ξ 0), un lacet positif contractile plus simple que dans PU (n, 1) et par exemple si le lacet ρ t , t ∈ S 1 , n'est pas contractile. C'est peu probable mais je n'en ai pas la preuve. The answer we provide is the following Corollary 2. The class in π 1 (Cont(S 2n+1 , ξ 0)) generated by the Reeb flow of α 0 is a non-trivial element of infinite order for n = 3. 1. Preliminaries 1.1. Contact structures. Definition 3. Let M 2n+1 be a smooth manifold. A codimension-1 regular distribution ξ is a contact distribution if there exists a 1-form α ∈ Ω 1 (M) such that ker α = ξ and α ∧ dα n is a volume form. The structure described above is known as a cooriented contact structure. Since the non-coorientable case is not considered in this article, we refer to a cooriented contact structure simply as a contact structure. The smooth manifold M will be assumed to be oriented. The contact structures to be considered will be positively cooriented, i.e. the induced orientation coincides with that prescribed on M. The definition is independent of the choice of 1-form α ′ = e f α, for any f ∈ C ∞ (M, R). Let Cont(M, ξ) = {s ∈ Diff(M) : ds * ξ = ξ} be the space of diffeomorphisms that preserve the contact structure. These diffeomorphisms are called contactomorphisms. The connected component of the identity of Cont(M, ξ) will be denoted by Cont 0 (M, ξ). C(M, ξ) will stand for the space of positive contact structures in M isotopic to ξ.
International Mathematics Research Notices, 2015
In this article we introduce the topological study of codimension-1 foliations which admit contac... more In this article we introduce the topological study of codimension-1 foliations which admit contact structures on the leaves. A parametric existence h-principle for foliated contact structures is provided for any cooriented foliation in a closed oriented 4-fold.
Proceedings - Mathematical Sciences, 2010
Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic str... more Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the "almost" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied.
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Papers by Francisco Presas