Daniel Alvarez-Gavela

The main result of this paper is the uniqueness of local arboreal models, defined as the closure of the class of smooth germs of Lagrangian submanifolds under the operation of taking iterated transverse Liouville cones. A parametric version implies that the space of germs of symplectomorphisms that preserve the local model is weakly homotopy equivalent to the space of automorphisms of the corresponding signed rooted tree. Hence the local symplectic topology around a canonical model reduces to combinatorics, even parametrically. This paper can be read independently, but it is part of a series of papers by the authors on the arborealization program.

We prove an ‘h‐principle without pre‐conditions’ for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result states that such tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non‐smooth points, called Lagrangian ridges, modeled on the corner {p=|q|}⊂R2$\lbrace p=|q|\rbrace \subset \mathbb {R}^2$ together with its products and stabilizations. This result plays an essential role in the arborealization program.

For each positive integer n, we give a geometric description of the stably trivial elements of the group p(n)U(n)/O-n. In particular, we show that all such elements admit representatives whose tangencies with respect to a fixed Lagrangian plane consist only of folds. By the h-principle for the simplification of caustics, this has the following consequence: if a Lagrangian distribution is stably trivial from the viewpoint of a Lagrangian homotopy sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian homotopy sphere so that its tangencies with respect to the Lagrangian distribution are only of fold type. Thus the stable triviality of the Lagrangian distribution, which is a necessary condition for the simplification of caustics to be possible, is also sufficient. We give applications of this result to the arborealization program and to the study of nearby Lagrangian homotopy spheres.

We introduce a Legendrian invariant built out of the Turaev torsion of generating families. This invariant is defined for a certain class of Legendrian submanifolds of 1-jet spaces, which we call of Euler type. We use our invariant to study mesh Legendrians: a family of 2-dimensional Euler type Legendrian links whose linking pattern is determined by a bicolored trivalent ribbon graph. The Turaev torsion of mesh Legendrians is related to a certain monodromy of handle slides, which we compute in terms of the combinatorics of the graph. As an application, we exhibit pairs of Legendrian links in the 1-jet
space of any orientable closed surface which are formally equivalent, cannot be distinguished by any natural Legendrian invariant, yet are not Legendrian isotopic. These examples appeared in a different guise in the work of the second author with J. Klein on pictures for $K_3$ and the higher Reidemeister torsion of circle bundles.

We establish a full h-principle (close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy.

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the h-principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working "one pure partial derivative at a time" is central to the discussion. We give applications of our results to flexible symplectic and contact topology.

En este trabajo se abordan algunas cuestiones relativas a la ecuación de la superficie mínima y su problema de Dirichlet asociado. Seguiremos los pasos de Giusti, caracterizando las condiciones necesarias y suficientes para la existencia y unicidad de soluciones a dicho problema, obteniendo sencillos criterios geométricos sobre la frontera del dominio en cuestión. Omnipresente en el reino de las superficies mínimas, el concepto de curvatura media jugará un papel protagonista. Por otra parte se presentarán diversas estimaciones y acotaciones que pondrán de manifiesto la importancia del principio del máximo, siendo aquí clave la convexidad del funcional del área.