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Ben Warhurst

Publications and source records attributed to Ben Warhurst.

14 recordsLinked to original sources
arXiv2026

Harmonic morphisms of sub-Riemannian Lie groups

We prove that harmonic morphisms between sub-Riemannian Lie groups are smooth, are symmetries of the sub-Riemannian Laplacian and are conformal submersions that satisfy a particular PDE. Moreover, we give some partial results for harmonic morphisms beyond Lie groups. We show the existence of harmonic coordinates for harmonically homogeneous spaces, and that smooth harmonic morphisms are symmetries of the Laplacian in all sub-Riemannian manifolds. Finally, we compute the first variation of the horizontal energy along a harmonic morphism of sub-Riemannian Lie groups: it is given by a linear form on the Lie algebra of the target, the modular mismatch, which vanishes for Riemannian and Carnot targets, but surprisingly not in general. Therefore, unlike in the Riemannian case, harmonic morphisms of sub-Riemannian Lie groups need not be harmonic maps.

math.DG↗
arXiv2024

Equivalence of sub-Laplacian on Polarized groups

We characterize smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians. We show they are sub-Riemannian conformal submersions. Our work clarifies the analysis initiated on Carnot groups in \cite{MR2363343}. In particular, we show that the sub-Laplacian in a Carnot group determines the sub-Riemannian structure.

math.DG↗
arXiv2024

Harmonic Curves From Euclidean Domains to Heisenberg Group H1

We define and study the harmonic curves on domains in $\mathbb{R}^n$ into the first Heisenberg group $\mathbb{H}^1$. These are the $C^2$-regular mappings which are critical points of the second Dirichlet energy and satisfy the weak isotropicity condition. We investigate the geometry of such curves including the comparison and maximum principles, the Harnack inequalities, the Liouville theorems, the existence results, the Phragm\`en-Lindel\"of theorem, as well as the three spheres theorem.

math.AP↗
arXiv2021

Schwarzians on the Heisenberg group

We study various notions of the Schwarzian derivative for contact mappings in the Heisenberg group $\mathbb{H}_1$ and introduce two definitions: (1) the CR Schwarzian derivative based on the conformal connection approach studied by Osgood and Stowe and, recently, by Son; (2) the classical type Schwarzian refering to the well-known complex analytic definition. In particular, we take into consideration the effect of conformal rigidity and the limitations it imposes. Moreover, we study the kernels of both Schwarzians and the cocycle conditions. Our auxiliary results include a characterization of the contact conformal vector fields. Inspired by ideas of Chuaqui--Duren--Osgood, Hernández, Martín and Venegas, we introduce the Preschwarzian for mappings in $\mathbb{H}_1$. Furthermore, we study results in the theory of subelliptic PDEs for the horizontal Jacobian and related differential expressions for harmonic mappings and the gradient harmonic mappings, the latter notion introduced here in the setting of $\mathbb{H}_1$.

math.AP↗
arXiv2019

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in $\mathbb{H}_1$. The theorems are discussed for the sub-Riemannian and the Korányi distances. This extends results due to Astala--Gehring, Astala--Koskela, Koskela and Bonk--Koskela--Rohde.

math.MG↗
arXiv2018

Puncture repair on metric measure spaces

Motivated by recent interest concerning "puncture repair" in the conformal geometry of compact Riemannian manifolds, a brief exposition on generalisation to the setting of quasiconformal mappings on certain metric measure spaces is presented as well as a brief outline on removability of porous sets.

math.CV↗
arXiv2017

Mean value property and harmonicity on Carnot-Carathéodory groups

We study strongly harmonic functions in Carnot-Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that spherical harmonic polynomials in $\mathbb{H}_1$ are both strongly harmonic and satisfy the sub-Laplace equation. Our presentation is illustrated by examples.

math.AP↗
arXiv2015

Prime ends in the Heisenberg group $\mathbb{H}_{1}$ and the boundary behavior of quasiconformal mappings

We investigate prime ends in the Heisenberg group $\mathbb{H}_{1}$ extending Näkki's construction for collared domains in Euclidean spaces. The corresponding class of domains is defined via uniform domains and the Loewner property. Using prime ends we show the counterpart of Caratheodory's extension theorem for quasiconformal mappings, the Koebe theorem on arcwise limits, the Lindelöf theorem for principal points and the Tsuji theorem.

math.MG↗
arXiv2015

Invariants and Infinitesimal Transformations for Contact Sub-Lorentzian Structures on 3-Dimensional Manifolds

In this article we develop some elementary aspects of a theory of symmetry in sub-Lorentzian geometry. First of all we construct invariants characterizing isometric classes of sub-Lorentzian contact 3 manifolds. Next we characterize vector fields which generate isometric and conformal symmetries in general sub-Lorentzian manifolds. We then focus attention back to the case where the underlying manifold is a contact 3 manifold and more specifically when the manifold is also a Lie group and the structure is left-invariant.

math.DG↗
arXiv2013

Ultrarigid tangents of sub-Riemannian nilpotent groups

We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In particular, such spaces are not locally bi-Lipschitz homeomorphic. The result is based on the study of Carnot groups that are rigid in the sense that their only quasiconformal maps are the translations and the dilations.

math.MG↗
arXiv2010

A Liouville type theorem for Carnot groups

L. Capogna and M. Cowling showed that if $ϕ$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $ϕ$ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that $ϕ$ is in fact $C^\infty$. As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2. In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory.

math.AP↗