## Research

### Papers and preprints

**Pablo Angulo, Daniel Faraco, Luis Guijarro:**

*Sufficient conditions for the existence of limiting Carleman weights*

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
$LCW$.

**Pablo Angulo:**

*Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds*

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 \(3\).**Pablo Angulo:**

*On the set of metrics without local limiting Carleman weights*

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.

**Pablo Angulo, Daniel Faraco, Luis Guijarro and Alberto Ruiz:**

*Obstructions to the existence of limiting Carleman weights*

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.

**Pablo Angulo:**

*Cut and conjugate points of the exponential map, with applications*

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**.**Pablo Angulo, Luis Guijarro, Gerard Walschap**

*Twisted submersions in nonnegative sectional curvature*

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.**Pablo Angulo, Luis Guijarro**

*Balanced split sets and Hamilton-Jacobi equations*

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.

**Pablo Angulo, Luis Guijarro**

*Cut and singular loci up to codimension 3*

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 \(n-3\).