Gaven Martin | Massey University (original) (raw)

Papers by Gaven Martin

$Research paper thumbnail of The Teichm\"uller problem for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup></mrow><annotation encoding="application/x-tex">L^p</annotation></semantics></math>Lp-means of distortion$

arXiv (Cornell University), Jul 15, 2021

arXiv (Cornell University), Jul 2, 2020

Elements f of finite order in the isometry group of hyperbolic threespace H 3 have a hyperbolic l... more Elements f of finite order in the isometry group of hyperbolic threespace H 3 have a hyperbolic line as a fixed point set, this line is the axis of f. The possible hyperbolic distances between axes of elements of order p and q, not both two, among all discrete subgroups Γ of Isom + (H 3) has an initial discrete spectrum 0 = δ 0 < δ 1 < δ 2 <. .. < δ ∞ , * This research was supported in part by a grant from the NZ Marsden Fund.

Complex Analysis and Operator Theory, Aug 2, 2023

The generalised Hopf equation is the first order nonlinear equation defined on a planar domain ⊂ ... more The generalised Hopf equation is the first order nonlinear equation defined on a planar domain ⊂ C, with data a holomorphic function and η ≥ 1 a positive weight on , h w h w η(w) =. The Hopf equation is the special case η(w) =η(h(w)) and reflects that h is harmonic with respect to the conformal metric η(z)|dz|, usually η is the hyperbolic metric. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.

arXiv (Cornell University), Jul 6, 2014

We give a new application of the theory of holomorphic motions to the study the distortion of lev... more We give a new application of the theory of holomorphic motions to the study the distortion of level lines of harmonic functions and stream lines of ideal planar fluid flow. In various settings, we show they are in fact quasilines-the quasiconformal images of the real line. These methods also provide quite explicit global estimates on the geometry of these curves.

arXiv (Cornell University), Nov 4, 2013

This chapter presents a survey of the many and various elements of the modern higher-dimensional ... more This chapter presents a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts. In the sequel (with Bruce Palka) we will give a more detailed account of the basic techniques and how they are used with a view to providing tools for researchers who may come in contact with higher-dimensional quasiconformal mappings from their own area.

$Research paper thumbnail of The <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(4,p)</annotation></semantics></math>(4,p)-arithmetic hyperbolic lattices, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p\geq 2</annotation></semantics></math>p≥2, in three dimensions$

arXiv (Cornell University), Jun 28, 2022

We identify the finitely many arithmetic lattices Γ in the orientation preserving isometry group ... more We identify the finitely many arithmetic lattices Γ in the orientation preserving isometry group of hyperbolic 3-space H 3 generated by an element of order 4 and and element of order p ≥ 2. Thus Γ has a presentation of the form Γ ∼ = f, g : f 4 = g p = w(f, g) = • • • = 1 We find that necessarily p ∈ {2, 3, 4, 5, 6, ∞}, where p = ∞ denotes that g is a parabolic element, the total degree of the invariant trace field kΓ = Q({tr 2 (h) : h ∈ Γ}) is at most 4, and each orbifold is either a two bridge link of slope r/s surgered with (4, 0), (p, 0) Dehn surgery (possibly a two bridge knot if p = 4) or a Heckoid group with slope r/s and w(f, g) = (w r/s) r with r ∈ {1, 2, 3, 4}. We give a discrete and faithful representation in P SL(2, C) for each group and identify the associated number theoretic data.

$Research paper thumbnail of On the Conformal Energy of Quasisymmetric and Quasim\"obius Mappings$

arXiv (Cornell University), May 21, 2023

This article identifies the conformal energy (or mean distortion) of extremal mappings of finite ... more 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 : S → S is * Research of both authors supported by the NZ Marsden Fund,

$Research paper thumbnail of Random Lattices, Punctured Tori and the Teichm\"uller distribution$

arXiv (Cornell University), Jul 29, 2018

The moduli space of lattices of C is a Riemann surface of finite hyperbolic area with the square ... more The moduli space of lattices of C is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teichmüller space to the central "square" punctured torus in the moduli space of punctured tori. There are singularities in this p.d.f. arising from the topology of the moduli space. We also consider the statistics of the distance in Teichmüller space to the rectangular punctured tori and the p.d.f and expected distortion of the extremal quasiconformal mappings.

Neural Processing Letters, May 1, 2020

Current multi-view clustering algorithms use multistage strategies to conduct clustering, or requ... more Current multi-view clustering algorithms use multistage strategies to conduct clustering, or require cluster number or similarity matrix prior, or suffer influence of irrelevant features and outliers. In this paper, we propose a Joint Robust Multi-view (JRM) spectral clustering algorithm that considers information from all views of the multi-view dataset to conduct clustering and solves the issues, such as initialization, cluster number determination, similarity measure, feature selection, and outlier reduction around clustering, in a unified way. The optimal performance could be reached when all views are considered and the separated stages are combined in a unified way. Experiments have been performed on six real-world benchmark datasets and our proposed JRM algorithm outperforms the comparison clustering algorithms in terms of two evaluation metrics for clustering algorithms including accuracy and purity.

Proceedings of The London Mathematical Society, Sep 1, 1987

arXiv (Cornell University), Apr 28, 2023

The generalised Hopf equation is the first order nonlinear equation with data Φ a holomorphic fun... more The generalised Hopf equation is the first order nonlinear equation with data Φ a holomorphic functions and η ≥ 1 a positive weight, h w h w η(w) = Φ. The Hopf equation is the special case η(w) =η(h(w)) and reflects that h is harmonic with respect to the conformal metric η(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.

arXiv (Cornell University), Feb 2, 2023

We consider the convexity properties of distortion functionals, particularly the linear distortio... more We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, n ≥ 3. The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if {f n } ∞ n=1 is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f , then this limit function is also K-quasiconformal.Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion H(f n)), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f , there is a sequence {f n } ∞ n=1 with f n → f locally uniformly and with lim sup n→∞ H(f n) < H(f). Our main result shows this is true for affine mappings. Addressing conjectures of F.W. Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds : for each α < √ 2 there is f n → f locally uniformly with f affine and α lim sup n→∞ H(f n) < H(f) We conjecture √ 2 to be best possible.

arXiv (Cornell University), Jul 12, 2022

For an arbitrary convex function Ψ : [1, ∞) → [1, ∞), we consider uniqueness in the following two... more For an arbitrary convex function Ψ : [1, ∞) → [1, ∞), 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 D Ψ(K(z, f)) dz : f :D →D a homeomorphism in W 1,1 0 (D) + f 0. Here the data f 0 :D →D is a homeomorphism of finite distortion with D Ψ(K(z, f 0)) dz < ∞-a barrier. Next, given two homeomorphic Riemann surfaces R and S and data f 0 : R → S a diffeomorphism.

arXiv (Cornell University), Nov 28, 2016

Frederick William Gehring was a hugely influential mathematician who spent most of his career at ... more Frederick William Gehring was a hugely influential mathematician who spent most of his career at the University of Michigan-appointed in 1955 and as the T H Hildebrandt Distinguished University Professor from 1987. Gehring's major research contributions were to Geometric Function Theory, particularly in higher dimensions R n , n ≥ 3. This field he developed in close coordination with colleagues, primarily in Finland, over three decades 1960-1990. Gehring's seminal work drove this field forward initiating important connections with geometry and nonlinear partial differential equations, while addressing and solving major problems. During his career Gehring received many honours from the international mathematical community.

Contemporary mathematics, 2005

arXiv (Cornell University), Oct 17, 2013

In 1936 H. Lewy showed that the Jacobian determinant of a harmonic homeomorphism between planar d... more In 1936 H. Lewy showed that the Jacobian determinant of a harmonic homeomorphism between planar domains does not vanish and thus the map is a diffeomorphism. This built on the earlier existence results of Radó and Kneser. R. Shoen and S.T. Yau generalised this result to degree 1 harmonic mappings between closed Riemann surfaces. Here we give a new approach that establishes all these results in complete generality.

arXiv (Cornell University), Jan 3, 2018

In earlier work we introduced geometrically natural probability measures on the group of all Möbi... more In earlier work we introduced geometrically natural probability measures on the group of all Möbius transformations in order to study "random" groups of Möbius transformations, random surfaces, and in particular random two-generator groups, that is groups where the generators are selected randomly, with a view to estimating the likely-hood that such groups are discrete and then to make calculations of the expectation of their associated parameters, geometry and topology. In this paper we continue that study and identify the precise probability that a Fuchsian group generated by two parabolic Möbius transformations is discrete, and give estimates for the case of Kleinian groups generated by a pair of random parabolic elements which we support with a computational investigation into of the Riley slice as identified by Bowditch's condition, and establish rigorous bounds.

Bloch-waves in 1D periodic lattices are typically constructed based on the transfer-matrix approa... more Bloch-waves in 1D periodic lattices are typically constructed based on the transfer-matrix approach, with a complete system of solutions of the Cauchy problem on a period. This approach fails for the multi-dimensional Schrödinger equations on periodic lattices, because the Cauchy problem is ill-posed for the associated elliptic partial differential equations. In our previous work [8] we suggested a different procedure for the calculation of the Bloch functions for the 2D Schrödinger equation based on the Dirichlet-to-Neumann map substituted for the transfer-matrix. In this paper we suggest a method of calculation of the dispersion function and Bloch waves of quasi-2D periodic lattices, in particular of a quasi-2D sandwich, based on construction of a fitted solvable model.