Quasiconformal mappings, with applications to differential equations, function theory and topology

Bers, Lipman

doi:10.1090/s0002-9904-1977-14390-5

Cited by 53 publications

(21 citation statements)

References 43 publications

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“…Explicit computation of the accessory parameter then would give the desired relationship and also seemed to give another model for the moduli space. Indeed, in a modern version [3], the accessory parameter which arises in the quasi-Fuchsian uniformization of the punctured torus is a complex analytic parameter for the moduli space. Accessory parameters are interesting in their own right.…”

Section: Introductionmentioning

confidence: 99%

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Moduli of punctured tori and the accessory parameter of Lamé’s equation

Keen¹,

Rauch²,

Vasquez³

1979

Trans. Amer. Math. Soc.

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Abstract.To solve the problems of uniformization and moduli for Riemann surfaces, covering spaces and covering mappings must be constructed, and the parameters on which they depend must be determined. When the Riemann surface is a punctured torus this can be done quite explicitly in several ways. The covering mappings are related by an ordinary differential equation, the Lamé equation. There is a constant in this equation which is called the "accessory parameter". In this paper we study the behavior of this accessory parameter in two ways. First, we use Hill's method to obtain implicit relationships among the moduli of the different uniformizations and the accessory parameter. We prove that the accessory parameter is not suitable as a modulus-even locally. Then we use a computer and numerical techniques to determine more explicitly the character of the singularities of the accessory parameter. 0. Introduction. In the classical theory of uniformization of tori there are two distinct models for the space of moduli: the complex analytic t (the period ratio) in the upper half plane and the real analytic trace parameters of Fricke for the Fuchsian uniformization of a punctured torus. Our original goal was to find an explicit relation between these models of Teichmüller space. They are related via an ordinary differential equation which is completely determined except for an unknown constant, called the "accessory parameter". Explicit computation of the accessory parameter then would give the desired relationship and also seemed to give another model for the moduli space. Indeed, in a modern version [3], the accessory parameter which arises in the quasi-Fuchsian uniformization of the punctured torus is a complex analytic parameter for the moduli space. Accessory parameters are interesting in their own right. There is an extensive discussion of accessory parameters in the classical literature; however, little of a concrete nature was known. We hope that what we have done here sheds some light on the general problem of accessory parameters.In this paper, we prove that the accessory parameter is not even a local parameter for Teichmüller space and we give qualitative information about it;

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Section: Introductionmentioning

confidence: 99%

“…Qualitative information about <p. This super-modular equivariance gives a great deal of information about the map <p: U^> *ÍF£; e.g., the unique fixed point in U of t -> -((1 + t)/t) is p = (-1 + V3 i)/2 = e(2tri/3). Hence <Kp) = (xx,yx, z,) is fixed by (x, v, z)->(v, z, x); and <p(p) = (3,3,3). Simi-larly <p(z) = (2V2 , 2V2 , 4).…”

mentioning

confidence: 97%

Moduli of punctured tori and the accessory parameter of Lamé’s equation

Keen¹,

Rauch²,

Vasquez³

1979

Trans. Amer. Math. Soc.

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“…It is fair to say that Teichmüller theory was traditionally a topic in complex analysis with a beautiful and extensive theory of Com(F ) ≈ Con(F ) based on Banach manifolds in the work of Ahlfors and Bers [20,21,68] and their school which makes precise our first definition of the Riemann moduli space. The definition can also be formulated in terms of algebraic geometry [33,51,58,89] though this is again rather involved.…”

Section: Triangles In Neutral Geometriesmentioning

confidence: 99%

Moduli spaces and macromolecules

Penner

2016

Bull. Amer. Math. Soc.

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Abstract. Techniques from moduli spaces are applied to biological macromolecules. The first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally. The second main result identifies up to homotopy the natural moduli space of several interacting RNA molecules with the Riemann moduli space of a surface with several boundary components in each fixed genus. Applications to RNA folding prediction are discussed. The mathematical and biological frameworks are surveyed and presented from first principles.

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“…We remark that these models are very relevant in practical applications since they are associated with the propagation of stable spatial solitons [20,21] We would like to emphasize that elliptic second order nonlinear equations of the form (4.4) possess several remarkable analytical and geometrical properties (see e.g. [19,22,23,24,25]). An interesting class of solutions of equation (4.4) is provided by the Beltrami equation which is well known and well studied in the theory of elliptic systems of PDEs and in the theory of quasiconformal mappings [24,25].…”

Section: Elliptic Intensity Lawsmentioning

confidence: 99%

High Frequency Integrable Regimes in Nonlocal Nonlinear Optics

Moro

Konopelchenko

2012

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We consider an integrable model which describes light beams propagating in nonlocal nonlinear media of Cole-Cole type. The model is derived as high frequency limit of both Maxwell equations and the nonlocal nonlinear Schrödinger equation. We demonstrate that for a general form of nonlinearity there exist selfguided light beams. In high frequency limit nonlocal perturbations can be seen as a class of phase deformation along one direction. We study in detail nonlocal perturbations described by the dispersionless Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction method based on symmetry constraints and by the quasiclassical∂−dressing method. Quasiclassical∂−dressing method reveals a connection between nonlocal nonlinear geometric optics and the theory of quasiconformal mappings of the plane.

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Quasiconformal mappings, with applications to differential equations, function theory and topology

Cited by 53 publications

References 43 publications

Moduli of punctured tori and the accessory parameter of Lamé’s equation

Moduli of punctured tori and the accessory parameter of Lamé’s equation

Moduli spaces and macromolecules

High Frequency Integrable Regimes in Nonlocal Nonlinear Optics

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