arXiv subjects
Publications and source records attributed to Gaven Martin.
Given $h_o: \mathbb{S} \to \mathbb{S}$, an orientation-preserving homeomorphism of the unit circle, we seek to establish very weak conditions so as to derive a modulus of continuity for $h_o$ and its inverse $h_o^{-1}$, then use these to derive various compactness properties for families of circle homeomorphisms and solutions to Beltrami equations.
We study the hyperbolically weighted distortion energies \[ \mathcal{I}_p(f)=\int_\mathbb{D} \frac{|μ_f|^p}{(1-|μ_f|^2)^{\max\{2,p\}/2}}\,dA_η\] for finite-distortion disk homeomorphisms with prescribed boundary values, together with the associated inverse problem in the monotone Sobolev class. For $1 2$. \
A circle homeomorphism $β:\mathbb{S}\to\mathbb{S}$ is Weil--Petersson if it has a quasiconformal disk extension whose Beltrami coefficient is square integrable for hyperbolic area. We minimise the hyperbolic anti-holomorphic energy \[ E_2(h)=\int_\mathbb{D}η(h)|h_{\bar w}|^2,dA, \quad\text{equivalently}\quad \int_\mathbb{D}\frac{|μ_f|^2}{1-|μ_f|^2},dA_η,\quad f=h^{-1}, \] over all monotone Sobolev maps with boundary values $β$. For Weil--Petersson data, the unique minimiser is a real analytic hyperbolic harmonic diffeomorphism, globally quasiconformal and asymptotically conformal, with an explicit dilatation bound in terms of any admissible barrier's energy. We prove that a circle homeomorphism is Weil--Petersson if and only if it admits a finite-distortion homeomorphic extension satisfying \[ \int_\mathbb{D}(\mathbb{K}_F-1),dA_η<\infty. \] For every $p>1$, the boundary values of a finite-distortion homeomorphism satisfying \[ \int_\mathbb{D}|μ_F|^p(1-|μ_F|^2)^{-p_*/2},dA_η<\infty \] lie in Guo's $p$-integrable Teichm"uller space $T^p$. A capacity estimate for boundary moduli for the quasisymmetric ratio yields a derivative-free characterisation of $T^p$, essentially Bishop's theorem when $p=2$. The minimum defines a M"obius-invariant boundary energy, quantitatively comparable to the weighted square norm of the minimiser's Hopf differential. For arbitrary quasisymmetric data, we prove Markovi'c's theorem (the Schoen conjecture) by approximation with our smooth quadratic minimisers. Mori's theorem and the mean-value property of a logarithm verify the averaged-distance hypothesis in Riestenberg--Smillie's interior comparison, giving explicit displacement, energy-density and dilatation bounds in terms of the quasisymmetry constant. The nonquadratic ($p\neq2$) theory is developed in Part~II.
We construct examples of uniformly quasiregular mappings (uqr) acting on a sphere and having Fatou set consisting of infinitely many components. In particular we construct a uqr mapping providing a higher dimensional counterpart for the polynomial $z \mapsto z^2 - 1$ acting on the extended complex plane $\hat{\mathbb{C}}$.
We consider minimisers in the homotopy class of a homeomorphism $f_0:R\to S$ between analytically finite Riemann surfaces with minimal $L^p$- conformal energy \[ \mathsf{E}_p(f:R,S)=\int_R \IK^p(z,f)\; dσ_R(z). \] The problem was first raised by Ahlfors in his celebrated proof of Teichmüller's theorem-the case $p=\infty$, but the existence, topological regularity and analytic regularity of these $L^p$ minimisers remained unknown for all $1<p<\infty$. Ahlfors established weak existence for $p\geq 2$. Here we prove that for all p, $1\leq p<\infty$, such minimisers exist, are unique and are diffeomorphisms. They are quasiconformal but not diffeomorphic at $p=\infty$.
Here we advance the study of boundary the value problem for extremal functions of mean distortion and the associated Teichmüller spaces interpolating between the classical examples of extremal quasiconformal mappings, and the more recent approach through harmonic mappings (of extreme Dirichlet energy). In this paper we focus on the Alhfors-Hopf differential \[ Φ=\mathcal{A}(\mathbb{K}(w,h))h_w\,\overline{h_{\overline{w}}}\, η(h), \] where $h=f^{-1}$ is the pseudo-inverse of an extremal mapping $f$ for the problem \[ \inf_{f:\mathbb{D}\to\mathbb{D}}\int_\mathbb{D} \mathcal{A}(\mathbb{K}(z,f)) \; dz, \quad\quad \mathbb{K}(z,f) = \frac{|f_z|^2+|f_{\overline{z}}|^2}{|f_z|^2-|f_{\overline{z}}|^2}. \] where the infimum is taken over those homeomorphisms of finite distortion $f:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ with $f|\mathbb{S}=f_0$, typically a quasisymmetric barrier function. The inner-variational equations, an analogue of the Euler-Lagrange equations, show $Φ$ is holomorphic at an extremal. Exploiting this Ahlfors-Hopf differential, we prove that an extreme point $f$ is a local diffeomorphism in $\mathbb{D}$, resolving some conjectures in [16].
We establish that the $p$-conformal energy, $p\geq 1$, defined by the $L^p$-norms of the distortion of Sobolev mappings, is a proper functional on the Teichmüller space of Riemann surfaces of a fixed genus. This result is an application of a result herein identifying explicitly both the unique extremal mappings of finite distortion between hyperbolic annuli of given modulus, and their $p$-conformal energy.
We consider minimisers of the $p$-exponential conformal energy for homeomorphisms $f:R \to S$ of finite distortion $\IK(z,f)$ between analytically finite Riemann surfaces in a fixed homotopy class $[f_0]$,\[ \mE_p(f:R,S)=\int_R \exp(p\IK(z,f))\; dσ(z). \] Homeomorphic minimisers exist should the barrier be a homeomorphism of finite energy, $\mE_p(f_0,R,S)<\infty$. In general this problem is not variational, however the Euler-Lagrange equations show the inverses $h=f^{-1}$ of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf differential, \[Φ=\exp(p\IK(w,h))\,h_w\overline{h_\wbar}\,dσ_R(h). \] From the Riemann-Roch theorem and an approximation technique, we show the variational equations hold for extremal mappings. We take this as a starting point for higher regularity to show that if $h:Ω\to\tildeΩ$ is a Sobolev homeomorphism between planar domains with holomorphic Ahlfors-Hopf differential, then $h$ is a diffeomorphism. It will follow that $h$ is harmonic in a metric induced by its own (smooth) distortion. We develop equations for the Beltrami coefficient of $h$, establishing a connection between degenerate elliptic non-linear Beltrami equations and these harmonic mappings. On the surface we conclude that minimisers $f_p\in [f_0]$ of $\mE_p(f:R,S)$ are diffeomorphisms and are unique stationary points. This now links two different approaches to Teichmüller theory; the classical theory of extremal quasiconformal maps and the harmonic mapping theory. As $p\to\infty$ we show $f_p\to f_\infty$ to recover the unique extremal quasiconformal mapping . This extremal quasiconformal mapping is not a diffeomorphism (unless it is conformal) and $f_p$ degenerates on a divisor. As $p\to0$ we recover the harmonic diffeomorphism in $[f_0]$ and Shoen-Yau's results.
We provide a brief overview of our upcoming work identifying all the thin Heckoid groups in $PSL(2,\mathbb{C})$. Here we give a complete list of the $55$ thin generalised triangle groups of slope $1/2$. This work was presented at the conference Computational Aspects of Thin Groups, IMSS, Singapore and presents an application of joint work initiated with Colin Maclachlan
We introduce a family of 3-variable "Farey polynomials" that are closely connected with the geometry and topology of $3$-manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-complex-dimensional deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank-two subgroups of PSL(2,C) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.
This article identifies the conformal energy (or mean distortion) of extremal mappings of finite distortion with a given quasisymmetric mapping of the circle as boundary data. The conformal energy of $g_o:\IS\to\IS$ is \begin{equation}\label{energy} \mathcal E(g_o)=-\frac{1}{4π^2}\iint_{\mathbb{S}\times \mathbb{S}}\log |g_o(ζ)-g_o(η)| \: dζd\barη < \infty \end{equation} We give explicit formulae for the conformal energy of circle homeomorphisms directly in terms of their data. As an example, if $g_o: \mathbb{S} \rightarrow \mathbb{S}$ is an $η$-quasi-Möbius self homeomorphism of the unit circle, then \begin{align*} \mathcal E(g_o) \leq \frac{1}π \int^{π/2}_0 \log η\big[ \cot^2(t/2)\big] \,\cos(t) \; dt \end{align*} This estimate is sharp. Additionally we show how a circle homeomorphism of finite conformal energy can be uniformly approximated on $\IS$ by mappings of strictly smaller energy.
The generalised Hopf equation is the first order nonlinear equation with data $Φ$ a holomorphic functions and $η\geq 1$ a positive weight, \[ h_w\,\overline{h_\wbar}\,η(w) = Φ.\] The Hopf equation is the special case $η(w)=\tildeη(h(w))$ and reflects that $h$ is harmonic with respect to the conformal metric $\sqrt{\tildeη(z)}|dz|$. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.
For an arbitrary convex function $Ψ:[1,\infty) \to [1,\infty)$, we consider uniqueness in the following two related extremal problems: Problem A boundary value problem: Establish the existence of, and describe the mapping $f$, achieving \[ \inf_f \Big\{ \int_{\Bbb D} Ψ({\Bbb K}(z,f))\; dz : f:\bar{\Bbb D} \to \bar{\Bbb D} \; \mbox{a homeomorphism in $W^{1,1}_{0}({\Bbb D})+f_0$} \Big\}. \] Here the data $f_0:\bar{\Bbb D} \to \bar{\Bbb D}$ is a homeomorphism of finite distortion with $\int_{\Bbb D} Ψ({\Bbb K}(z,f_0))\; dz<\infty$ -- a barrier. Next, given two homeomorphic Riemann surfaces $R$ and $S$ and data $f_0:R \to S$ a diffeomorphism. \noindent{\bf Problem B} {\em (extremal in homotopy class):} Establish the existence of, and describe the mapping $f$, achieving \[ \inf_f \Big\{ \int_R Ψ({\Bbb K}(z,f))\; \;dσ(z) : \mbox{$f$ a homeomorphism homotopic to $f_0$} \Big\}. \] There are two basic obstructions to existence and regularity. These are first, the existence of an Ahlfors-Hopf differential and second that the minimiser is a homeomorphism. When these restrictions are met (as they often can be) we show uniqueness is assured. These results are established through a generalisation the classical Reich-Strebel inequalities to this variational setting.
The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in $ \mathbb{C} $, and has a natural coordinate system (introduced by Linda Keen and Caroline Series in the early 1990s) which reflects the geometry of the underlying 3-manifold deformations. The Riley slice arises in the study of arithmetic Kleinian groups, the theory of two-bridge knots, the theory of Schottky groups, and the theory of hyperbolic 3-manifolds; because of its simplicity it provides an easy source of examples and deep questions related to these subjects. We give an introduction for the non-expert to the Riley slice and much of the related background material, assuming only graduate level complex analysis and topology; we review the history of and literature surrounding the Riley slice; and we announce some results of our own, extending the work of Keen and Series to the one complex dimensional moduli spaces of Kleinian groups isomorphic to $\mathbb{Z}_p*\mathbb{Z}_q$ acting on the Riemann sphere, $2\leq p,q \leq \infty$. The Riley slice is the case $p=q=\infty$ (i.e. two parabolic generators).
Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify `half-space' neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.
We survey a number of recent developments in geometric analysis as they pertain to the calculus of variations and extremal problems in geometric function theory following the NZMRI lectures given by the first author at those workshops in Napier in 1998 and 2005.
A two-generator Kleinian group $\langle f,g \rangle$ can be naturally associated with a discrete group $\langle f,ϕ\rangle$ with the generator $ϕ$ of order $2$ and where \begin{equation*} \langle f,ϕf ϕ^{-1} \rangle= \langle f,gfg^{-1} \rangle \subset \langle f,g\rangle, \quad [ \langle f,g f g^{-1} \rangle: \langle f,ϕ\rangle]=2 \end{equation*} This is useful in studying the geometry of Kleinian groups since $\langle f,g \rangle$ will be discrete only if $\langle f,ϕ\rangle$ is, and the moduli space of groups $\langle f,ϕ\rangle$ is one complex dimension less. This gives a necessary condition in a simpler space to determine the discreteness of $\langle f,g \rangle$. The dimension reduction here is realised by a projection of principal characters of two-generator Kleinian groups. In applications it is important to know that the image of the moduli space of Kleinian groups under this projection is closed and, among other results, we show how this follows from Jørgensen's results on algebraic convergence.
We consider Sobolev mappings $f\in W^{1,q}(Ω,\IC)$, $1<q<\infty$, between planar domains $Ω\subset \IC$. We analyse the Radon-Riesz property for convex functionals of the form \[f\mapsto \int_ΩΦ(|Df(z)|,J(z,f)) \; dz \] and show that under certain criteria, which hold in important cases, weak convergence in $W_{loc}^{1,q}(Ω)$ of (for instance) a minimising sequence can be improved to strong convergence. This finds important applications in the minimisation problems for mappings of finite distortion and the $L^p$ and $Exp$\,-Teichmüller theories.