scholarly journals ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS

2014 ◽  
Vol 56 (3) ◽  
pp. 681-689
Author(s):  
ZONGLIANG SUN
Keyword(s):  
Quasiconformal Mapping ◽  
Homotopy Class ◽  
Quadratic Differential ◽  
Riemann Surfaces ◽  
Simple Closed Curve ◽  
Quasiconformal Mappings ◽  
Closed Curve

AbstractIn this paper, we consider the question about length distortions under quasiconformal mappings with respect to quadratic differential metrics. More precisely, let X and Y be closed Riemann surfaces with genus at least 2, and f: X → Y being a K-quasiconformal mapping. Given two quadratic differential metrics |q1| and |q2| with unit areas on X and Y respectively, whether there exists a constant $\mathcal C$ depending only on K such that $\frac{1}{\mathcal C} l_{q_1} (\gamma) \leq l_{q_2} (f(\gamma)) \leq \mathcal C l_{q_1} (\gamma)$ holds for any simple closed curve γ ⊂ X. Here lqi(α) denotes the infimum of the lengths of curves in the homotopy class of α with respect to the metric |qi|, i = 1, 2. We give positive answers to this question, including the aspects that the desired constant ${\mathcal C}$ explicitly depends on q1, q2 and K, and that the constant $\mathcal C$ is universal for all the quantities involved.

scholarly journals Representing Homology Classes on Surfaces

1976 ◽  
Vol 19 (3) ◽  
pp. 373-374 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Unit Circle ◽  
Homotopy Class ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Integral Basis

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

scholarly journals Application of quasiconformal mapping methods to the investigation of the explosive processes impact on the environment

2019 ◽  
pp. 17-18
Author(s):  
Kateryna Mykolaiivna Malash ◽  
Andrii Yaroslavovych Bomba
Keyword(s):  
Numerical Methods ◽  
Mathematical Models ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Practical Application

The mathematical models used to study explosive processes are given. A class of problems investigating the influence of explosive processes on the environment by the quasiconformal mappings numerical methods are outlined and their practical application are described

A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc

1960 ◽  
Vol 12 ◽  
pp. 209-230 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
History Of ◽  
Homogeneous Plane ◽  
Plane Continuum

One of the unsolved problems of plane topology is the following:Question. What are the homogeneous bounded plane continua?A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

scholarly journals On a Theorem of Schwarz Type for Quasiconformal Mappings in Space

1967 ◽  
Vol 29 ◽  
pp. 19-30
Author(s):  
Kazuo Ikoma
Keyword(s):  
Conformal Mapping ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Space Domain ◽  
Moebius Transformation ◽  
Modulus Condition

A space ring R is defined as a domain whose complement in the Moebius space consists of two components. The modulus of R can be defined in variously different but essentially equivalent ways (see e.g. Gehring [3] and Krivov [5]), which is denoted by mod R. Following Gehring [2], we refer to a homeomorphism y(x) of a space domain D as a k-quasiconformal mapping, if the modulus conditionis satisfied for all bounded rings R with their closure , where y(R) denotes the image of R by y = y(x). Then, it is evident that the inverse of a k-quasi-conformal mapping is itself k-quasiconformal and that a k1-quasiconformal mapping followed by a k2-quasiconformal one is k1k2-quasiconformal. It is also well known that the restriction of a Moebius transformation to a space domain is equivalent to a 1-quasiconformal mapping of its domain.

scholarly journals Handel’s fixed point theorem revisited

2012 ◽  
Vol 33 (5) ◽  
pp. 1584-1610
Author(s):  
JULIANA XAVIER
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unit Disk ◽  
Point Theorem ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Disk ◽  
Oriented Cycle

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

A Census of Planar Triangulations

1962 ◽  
Vol 14 ◽  
pp. 21-38 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Finite Number ◽  
Interior Point ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Region ◽  
Straight Segments

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.

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