arXiv research

The paper studies harmonic graphs in the Heisenberg group and their properties.

Study harmonicity of normal almost contact structures on Riemannian manifolds.

The study of harmonicity for almost contact metric structures was initiated by Vergara-Díaz and Wood and continued by González-Dávila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric structure, González-Dávila and the present author have characterised harm…

Study on harmonicity of maps between different types of almost contact metric manifolds.

An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are giv…

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonic…

We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric st…

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the a…

We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.

Spinors help study unique five-dimensional contact structures.

Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizin…

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.

A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.

Graphs describe contact surgery on 3-manifolds.

We will prove a Moser-type theorem for self-dual harmonic 2-forms on closed 4-manifolds, and use it to classify local forms on neighborhoods of singular circles on which the 2-form vanishes. Removing neighborhoods of the circles, we obtain a symplectic manifold with contact boundary - we show that the contact form on e…

Study Schwarzians in the Heisenberg group, introducing new definitions and characterizing contact conformal vector fields.

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using …

Study of harmonic functions on infinite penny graphs.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

We show that, under weak assumptions, the automorphism group of a ${\rm CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen's contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graph…

Study polynomial growth harmonic functions on infinite penny graphs.

Study shows how to create special metrics on 4-manifolds with certain spheres.

This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the $L^2$-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory sp…

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion--free and some types of them have to be locally flat. We prove that each symmetry given at a point with non--zero harmonic curvature is involutive. Finally we giv…

Finite graphs with specific curvature have limited harmonic functions and ends.

New spectral sequence for $\mathcal{K}$-manifolds, computing cohomology and harmonic forms.

We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except $\mathfrak{sl}_2$. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the $G_2$ case. We give a …

The study proves unique harmonic functions and combinatorial properties of vertex-transitive graphs.

We study compatible contact structures of fibered, positively-twisted graph multilinks in the 3-sphere and prove that the contact structure of such a multilink is tight if and only if the orientations of its link components are all consistent with or all opposite to the orientation of the fibers of the Seifert fibratio…

The contact graph of a CAT(0) cubical complex has unbounded structure and a Gaussian CLT for random walks.

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.

New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

Let $p$ be a real number greater number greater than one. Suppose that a graph $G$ of bounded degree is quasi-isometric with a Riemannian manifold $M$ with certain properties. Under these conditions we will show that the $p$-harmonic boundary of $G$ is homeomorphic to the $p$-harmonic boundary of $M$. We will also prov…

We characterize general pseudo-harmonic morphisms from a Riemannian manifold to a Hermitian manifold as pseudo horizontally weakly conformal maps with an additional property. We study to what extent we can (locally) describe these submersive pseudo-harmonic morphisms via the foliation given by the kernel of the associa…

Paper estimates Gaussian curvature of minimal graphs in a specific manifold.

The paper characterizes contact metric manifolds with specific solitons.

Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous func…

Every connected, weighted graph with non-negative curvature has exactly two ends.

We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legen…

Study Higgs sections and flat sections for nonlinear harmonic bundles.

A formula connects discrete harmonic surfaces to holomorphic functions.

We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, w…

Study Legendrian surfaces using N-graphs and flag moduli.

Study solutions and singularities of G2-structures flows on specific manifolds.

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and unif…

We introduce a novel harmonic analysis for functions defined on the vertices of a strongly connected directed graph of which the random walk operator is the cornerstone. As a first step, we consider the set of eigenvectors of the random walk operator as a non-orthogonal Fourier-type basis for functions over directed gr…

Graph-based methods for anomaly detection and semi-supervised learning.