Conformal Geometry and Quasiregular Mappings

Вуоринен, Матти

doi:10.1007/bfb0077904

Cited by 486 publications

(409 citation statements)

References 0 publications

Supporting

Mentioning

400

Contrasting

Unclassified

Order By: Relevance

“…Hence by Lemma 2 no(ri,O): no(ri,oo) for large j. Furthermore, it follows from [13,9.1]" and 9.2. (1)] that nn(ri,rb@)) : no(rit0) : nr(r;, oo) : nn(, j,rb(t)) for large j .…”

mentioning

confidence: 83%

See 1 more Smart Citation

On the behavior of quasiregular mappings in the neighborhood of an isolated singularity

Järvi¹

1990

Ann. Acad. Sci. Fenn. Ser. A I Math.

View full text Add to dashboard Cite

1. The purpose of the present study is to consider the behavior of quasimeromorphic mappings about an essential isolated singularity or, by contraposition, to seek conditions under which such maps possess a limit at a singular point. In Section 2 we will establish two Picard-type theorems for quasiregular mappings both of which generalize a recent result of Vuorinen [1a]. Furthermore, we will show that isolated singularities are removable for normal quasiregular mappings. Here normality means uniform continuity with respect to the quasihyperbolic metric. Section 3 is devoted to the value distribution of normal quasimeromorphic mappings. In Section 4 we will discuss the oscillation of quasimeromorphic mappings in the neighborhood of an essential singularity. This part of the work is related to Lehto's result concerning the growth of the spherical derivative of meromorphic functions near an isolated singularity [5].I wish to thank Matti Vuorinen for his useful comments.2. our notation and terminology will be mainly the same as in vuorinen's book [13], Accordingly, B"(*,r) denotes the ball centered at c € R" with rad.ius r while s"-'(r,r) is the sphere with the same center a^nd radius. For brevity, B"(r): B"(0,r), B": B"(1), sn-l(r) : ,5"-r(0,r), ,s'-1 : ,9"-,(1). The euclidean distance in R" is denoted by d(.,.), whereas g(.,.) refers to the chordal distance in R", the one-point compactificaiion of R", *rri"r, as usual is identified with the Riemann sphere S"('r"n*r, |) (see [18, p. a] Picard's theorem due to Rickman [11]: For every integer n ) 2 and each K ) 1 there exists a positive integer p : p(n,I() such that if .f : R" ---+ R"\{o1 ,. . . ,ap_t} is -K-quasiregular and a1, ... rdp-t are distinct points in R', then / is constant.

show abstract

“…Hence by Lemma 2 no(ri,O): no(ri,oo) for large j. Furthermore, it follows from [13,9.1]" and 9.2. (1)] that nn(ri,rb@)) : no(rit0) : nr(r;, oo) : nn(, j,rb(t)) for large j .…”

mentioning

confidence: 83%

“…our notation and terminology will be mainly the same as in vuorinen's book [13], Accordingly, B"(*,r) denotes the ball centered at c € R" with rad.ius r while s"-'(r,r) is the sphere with the same center a^nd radius. For brevity, B"(r): B"(0,r), B": B"(1), sn-l(r) : ,5"-r(0,r), ,s'-1 : ,9"-,(1).…”

mentioning

confidence: 99%

On the behavior of quasiregular mappings in the neighborhood of an isolated singularity

Järvi¹

1990

Ann. Acad. Sci. Fenn. Ser. A I Math.

View full text Add to dashboard Cite

show abstract

“…Nevertheless, our assumptions on the set E are in general slightly weaker than in the corresponding results of [5]. Earlier results concerning neighbourhood capacities can also be found in Väisälä [13] and Vuorinen [14,Sect. 6].…”

Section: Dist(x E) < T} and Call E T The (Open) T-neighbourhood Of Ementioning

confidence: 79%

“…Let us begin with a simple observation which gives a 'unversal' upper bound for the growth of neighbourhood capacities; this can be viewed as a generalization of a result of Vuorinen [14,Lemma 6.27], in which only the the case p = n was concerned.…”

Section: Upper Boundsmentioning

confidence: 99%

Neighbourhood capacities

Lehrbäck¹

2012

Ann. Acad. Sci. Fenn. Math.

View full text Add to dashboard Cite

Abstract. We study the behaviour of the p-capacity of a compact set E with respect to the t-neighbourhoods of E as t varies. We establish sharp upper and lower bounds for these capacities in terms of Minkowski and Hausdorff type contents of E, respectively, and our results hold in both Euclidean and more general metric spaces. In our lower bounds the porosity of the set E plays an important role, and it is shown by examples that an assumption like this is in general necessary. In addition, we present a self-contained approach to the theory of sets of zero capacity in metric spaces.

show abstract

“…The origin of the theory of modulus of curves in compact metric spaces must be found in the classical theory of quasiconformal maps in Euclidean spaces (see [42] or [44]). Quasiconformal maps are maps between homeomorphisms and bi-Lipschitz maps.…”

Section: Introduction Starting Pointmentioning

confidence: 99%

Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings

Clais

2016

Analysis and Geometry in Metric Spaces

View full text Add to dashboard Cite

Abstract:In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity of right-angled Fuchsian buildings.

show abstract

Conformal Geometry and Quasiregular Mappings

Cited by 486 publications

References 0 publications

On the behavior of quasiregular mappings in the neighborhood of an isolated singularity

On the behavior of quasiregular mappings in the neighborhood of an isolated singularity

Neighbourhood capacities

Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings

Contact Info

Product

Resources

About