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Álvaro del Pino

Publications and source records attributed to Álvaro del Pino.

At least 19 recordsLinked to original sources
arXiv2026

The h-principle fails for prelegendrians in corank 2 fat distributions

We investigate the $h$-principle problem for fat distributions. These are maximally non-integrable distributions with natural symplectisations and contactisations, that generalize contact distributions to higher corank. We focus on the corank-$2$ case, where we study a natural class of submanifolds, which we call prelegendrians. Their key feature is that they admit a canonical Legendrian lift to the contactisation. Our main results state that the $h$-principle fails for these submanifolds in all dimensions. To the best of our knowledge, this is the first example of rigidity in the study of maximally non-integrable distributions, outside of contact topology. First, we find an infinite family of $(2n+1)$-tori in the standard fat $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, with the following two properties: (1) They all represent the same formal prelegendrian class, (2) but they are not prelegendrian isotopic because they are distinguished by pseudoholomorphic curve invariants of their Legendrian lift. Secondly, we define the notion of prelegendrian stabilization in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$. This allows us to take an arbitrary prelegendrian and produce another one, in the same formal class, whose Legendrian lift is loose. In order to prove these results we also develop the fundamentals of the theory of prelegendrians. This includes: (1) introducing the notion of front projection in $(\mathbb{C}^{2n+1},\mathcal{D}_{\mathrm{std}})$, (2) proving that pseudoholomorphic curve invariants are robust under perturbations of the fat structure, allowing us to transport our results to non-standard fat structures, (3) introducing a zooming argument showing that any fat structure in dimension $6$ admits prelegendrians.

math.SG↗
arXiv2025

Engel CR submanifolds of $\mathbb{C}^3$

We give a sufficient condition for an $\mathbb{S}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into $\mathbb{C}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of $\mathbb{C}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg.

math.DG↗
arXiv2025

$b^k$-algebroids and the variety of foliation jets

We introduce and classify singular foliations of $b^{k+1}$-type, which formalize the properties of vector fields that are tangent to a submanifold $W \subset M$ to order $k$. When $W$ is a hypersurface, these structures are Lie algebroids generalizing the $b^{k+1}$-tangent bundles introduced by Scott. We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th order foliations and employ the holonomy invariant to show that these groupoids fiber over certain character stacks, yielding Riemann-Hilbert style classifications up to local isomorphism and isotopy. We also study the problem of extending a $k$-th order foliation to a $(k+1)$-st order foliation. We prove that this is obstructed by a characteristic class that arises as a section of a vector bundle over the relevant character stack.

math.DG↗
arXiv2025

Jiggling: an h-principle without homotopical assumptions

The jiggling lemma of Thurston shows that any triangulation can be jiggled (read: subdivided and then perturbed) to be in general position with respect to a distribution. Our main result is a generalization of Thurston's lemma. It states that piecewise smooth solutions of a given open and fiberwise dense differential relation $\mathcal{R} \subset J^1(E)$ of first order can be constructed by jiggling arbitrary sections of $E$. Our statement also holds in parametric and relative form. We understand this as an h-principle without homotopical assumptions for piecewise smooth solutions of $\mathcal{R}$.

math.GT↗
arXiv2025

Wrinkling and Haefliger structures

Wrinkling techniques, introduced by Eliashberg and Mishachev, are typically used to prove h-principles of the form: ``formal solutions of a partial differential relation $\mathcal{R}$ can be deformed to singular/wrinkled solutions''. What a wrinkled solution is depends on the context, but the overall idea is that it should be an object that fails to be a solution only due to the presence of mild/controlled singularities. Much earlier, Haefliger structures were introduced by Haefliger as singular analogues of foliations. Much like a foliation is locally modeled on a submersion, a Haefliger structure is modeled on an arbitrary map. This implies that Haefliger structures have better formal properties than foliations. For instance, they can be pulled back by arbitrary maps and admit a classifying space. In [12], the second and third authors generalized the \emph{wrinkled embeddings} of Eliashberg and Mishachev to arbitrary order. This paper can be regarded as a sequel in which we deal instead with generalizations of \emph{wrinkled submersions}. The main messages are that: 1) Haefliger structures provide a nice conceptual framework in which general wrinkling statements can be made. 2) Wrinkling can be interpreted as holonomic approximation into the étale space of solutions of the relation $\mathcal{R}$. These statements imply connectivity statements relating (1) $\mathcal{R}$ to its étale space of solutions and (2) the classifying space for foliations with transverse $\mathcal{R}$-geometry to its formal counterpart.

math.GT↗
arXiv2023

Jets of foliations and $b^k$-algebroids

In this article, we introduce and study singular foliations of $b^k$-type. These singular foliations formalize the properties of vector fields that are tangent to order $k$ along a submanifold $W \subset M$. Our first result is a classification of these foliations, relating them to geometric structures defined in a formal neighborhood of the submanifold, such as jets of distributions that are involutive up to order $k-1$. When $W$ is a hypersurface, singular foliations of $b^k$-type are Lie algebroids. In this particular case, they are generalizations of the $b^k$-tangent bundles introduced by Scott. Indeed, they are always locally isomorphic to $b^k$-tangent bundles, but globally such an isomorphism is obstructed by a holonomy invariant. Our second main result is a Riemann-Hilbert-style classification of singular foliations of $b^k$-type in terms of holonomy representations. In this paper, we study singular foliations of $b^k$-type from several different perspectives. In particular: (1) We study the problem of extending a $k$-th-order foliation to a $(k+1)$-th order foliation and prove that this is obstructed by a characteristic class. (2) When $W$ is a hypersurface, we give a detailed study of algebroid differential forms and extend Scott's calculation of the cohomology. (3) We study algebroid symplectic forms in terms of the geometric structures induced on $W$. In particular, we find that there is a close relationship between the above obstruction class for extensions and the symplectic variation of the symplectic foliation induced on $W$.

math.DG↗
arXiv2022

Regularisation of Lie algebroids and Applications

We describe a procedure, called regularisation, that allows us to study geometric structures on Lie algebroids via foliated geometric structures on a manifold of higher dimension. This procedure applies to various classes of Lie algebroids; namely, those whose singularities are of b^k, complex-log, or elliptic type, possibly with self-crossings. One of our main applications is a proof of the Weinstein conjecture for overtwisted b^k-contact structures. This was proven by Miranda-Oms using a certain technical hypothesis. Our approach avoids this assumption by reducing the proof to the foliated setting. As a by-product, we also prove the Weinstein conjecture for other Lie algebroids. Along the way we also introduce tangent distributions, i.e. subbundles of Lie algebroids, as interesting objects of study and present a number of constructions for them.

math.DG↗
arXiv2022

Classification of tangent and transverse knots in bracket-generating distributions

Consider a manifold, of dimension greater than 3, equipped with a bracket-generating distribution. In this article we prove complete h-principles for embedded regular horizontal curves and for embedded transverse curves. These results contrast with the 3-dimensional contact case, where the full h-principle for transverse/legendrian knots is known not to hold. We also prove analogous statements for immersions, with no assumptions on the ambient dimension.

math.DG↗
arXiv2021

Convex integration with avoidance and hyperbolic (4,6) distributions

This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration.

math.DG↗
arXiv2021

Wrinkling h-principles for integral submanifolds of jet spaces

Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding to show that any tangential homotopy can be approximated by a homotopy of topological embeddings with mild singularities. This concept plays an important role in Contact Topology: The loose legendrian h-principle of E. Murphy relies on wrinkled embeddings to manipulate the legendrian front. Similarly, the simplification of legendrian front singularities was proven by D. Álvarez-Gavela by defining the notion of wrinkled legendrian. This paper and its sequel generalise these ideas to general jet spaces. The main theorem in the present paper proves the analogue of the result by Eliashberg and Mishachev: Any homotopy of the r-order differential information of an embedding can be approximated by a homotopy of embeddings with wrinkle-type singularities (of order r). The local version of the previous statement, which is of independent interest, says that the holonomic approximation theorem holds over closed manifolds if, instead of sections, we consider multi-valued sections with simple singularities.

math.DG↗
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

Introducing sub-Riemannian and sub-Finsler Billiards

We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-defined at singular points of the boundary distribution, the appearance of gliding and creeping orbits, and the behavior of reflections at wavefronts. We then study some concrete tables in 3-dimensional euclidean space endowed with the standard contact structure. These can be interpreted as planar magnetic billiards, of varying magnetic strength, for which the magnetic strength may change under reflection. For each table we provide various results regarding periodic trajectories, gliding orbits, and creeping orbits.

math.DG↗
arXiv2020

Microflexiblity and local integrability of horizontal curves

Let $ξ$ be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of Control Theory) has infinite codimension within the space of germs of smooth curves tangent to $ξ$. We formalise this as an asymptotic statement about finite jets of tangent curves. This solves, in the analytic setting, a conjecture of Y. Eliashberg and N.M. Mishachev regarding an earlier claim by M. Gromov about the microflexibility of the tangency condition. From these statements it follows, by an argument due to M. Gromov, that the $h$-principle holds for maps and immersions transverse to $ξ$.

math.DG↗
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

The Engel-Lutz twist and overtwisted Engel structures

We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in 3-dimensional Contact Topology. This notion allows us to define what an Engel overtwisted disc is, and to prove a complete h-principle for overtwisted Engel structures with fixed overtwisted disc.

math.SG↗
arXiv2017

Classification of Engel Knots

For any Engel 4-fold, we show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the orbits of the Engel kernel. This is a relative, parametric and close h-principle.

math.GT↗
arXiv2017

On the classification of prolongations up to Engel homotopy

In [CPPP] it was shown that Engel structures satisfy an existence $h$-principle, and the question of whether a full $h$-principle holds was left open. In this note we address the classification problem, up to Engel deformation, of Cartan and Lorentz prolongations. We show that it reduces to their formal data as soon as the turning number is large enough. Somewhat separately, we study the homotopy type of the space of Cartan prolongations, describing completely its connected components in the overtwisted case.

math.SG↗