Pablo Angulo en la Universidad Autónoma de Madrid

arXiv In 1411.4887, we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based upon the Cotton-York tensor in dimension $3$ and the Weyl tensor in dimension $4$. In this paper, we find further necessary conditions for the existence of local LCWs that are often sufficient. For a manifold of dimension $3$ or $4$, we classify the possible Cotton-York, or Weyl tensors, and provide a mechanism to find out whether the manifold admits local LCW for each type of tensor. In particular, we show that a product of two surfaces admits a LCW if and only if at least one of the two surfaces is of revolution. This provides an example of a manifold satisfying the eigenflag condition of \cite{AFGR} but not admitting $LCW$.

arXiv We present a new strategy for proving the Ambrose conjecture, a global version of the Cartan local lemma. We introduce the concepts of linking curves, unequivocal sets and sutured manifolds, and use them to show that any sutured manifold satisfies the Ambrose conjecture. We then prove that the set of sutured Riemmanian manifolds contains a residual set of the metrics on a given smooth manifold of dimension \(3\).

arXiv In the paper AFGR it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper LS it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This note is a continuation of AFGR, in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights \emph{at any point} is open and dense.

arXiv We give a necessary condition for a Riemannian manifold to admit limiting Carleman weights in terms of its Weyl tensor (in dimensions 4 and higher), or its Cotton-York tensor in dimension 3. As an application we provide explicit examples of manifolds without limiting Carleman weights and show that the set of such metrics on a given manifold contains an open and dense set.

arXiv Entrada en Teseo The goal of this thesis is to study the singularities of the exponential map of \emph{Riemannian and Finsler manifolds} (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton, with applications to differential geometry, control theory, statistics, image processing...), to improve existing results about its structure, to look at it in new ways, and to derive applications to the Ambrose conjecture and the Hamilton-Jacobi equations.

arXiv Arch. Mat. In Wil, B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for Riemannian submersions from nonnegatively curved spaces even without the strict positive curvature assumption, and irrespective of the particular metric.

arXiv Calc. Var. PDE We study the singular set of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of \(u\). In a previous paper, we proved that the singular set is what we called a balanced split locus. In this paper, we find and classify all balanced split loci, identifying the cases where the only balanced split locus is the singular locus, and the cases where this does not hold. This clarifies the relationship between viscosity solutions and the classical approach of characteristics, providing equations for the singular set. Along the way, we prove more structure results about the singular sets.

arXiv Ann. Inst We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension \(n-2\) is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension \(n-3\).