Article
On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry
10/2011;
Source: arXiv
 Citations (13)
 Cited In (0)

Article: Quasiregular mapping and cohomology
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ABSTRACT: In a series of deep investigations S. Rickman studied the question of existence of a nonconstant quasiregular mapping f:ℝ n →Y, n≥2, for target spaces of the form ℝ n ∖E where E is a finite set. In particular, he proved a counterpart of Picard’s theorem [J. Anal. Math. 37, 100–117 (1980; Zbl 0451.30012)] in this context and also proved that this result is qualitatively best possible for n=3 [Acta Math. 154, 195–242 (1985; Zbl 0617.30024)]. See also the survey [Quasiconformal space mappings, Collect. Surv. 19601990, Lect. Notes Math. 1508, 93–103 (1992; Zbl 0764.30017)] of Rickman. The RickmanPicard theorem was an important landmark for the development of quasiregular mappings and has also found many applications in the works of his colleagues and students, e.g. in P. Järvi and M. Vuorinen, J. Reine Angew. Math. 424, 31–45 (1992; Zbl 0733.30017), J. Jormakka, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 69. Helsinki: Suomalainen Tiedeakatemia; Univ. of Helsinki, Faculty of Science (1988; Zbl 0662.57007), J. Kankaanpää, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 110. Helsinki: Suomalainen Tiedeakatemia; Univ. of Helsinki, Faculty of Science (1997; Zbl 0909.30014), K. Peltonen, Ann. Acad. Sci. Fenn., Ser. A I. Math., Diss. 85. Helsinki: Suomalainen Tiedeakatemia; Univ. of Helsinki, Faculty of Science (1992; Zbl 0757.53024). One of the chief themes is to study the case when the target space Y is a suitable manifold. Rickman’s joint work with I. Holopainen [Andreian Cazacu, Cabiria (ed.) et al., Analysis and topology. A volume dedicated to the memory of S. Stoilow. Singapore: World Scientific, 315–326 (1998; Zbl 0945.30022)] has developped these ideas further. The authors of the paper under review penetrate deeper into this difficult territory. Their main result is the following theorem. Theorem Let N be a closed, connected and oriented Riemannian nmanifold, n≥2· If there exists a nonconstant Kquasiregular mapping f:ℝ n →N, then dimH * (N)≤C(n,K) where dimH * (N) is the dimension of the de Rham cohomology ring H n (N) of N, and C(n,K) is a constant depending only on n and K· The tools the authors use include methods from Rickman’s value distribution theory and also some results of T. Iwaniec and his coauthors [e.g. Arch. Ration. Mech. Anal. 125, No. 1, 25–79 (1993; Zbl 0793.58002)] on differential forms.Acta Mathematica 08/2001; 186(2):219238. · 2.71 Impact Factor 
Article: Quasiconformal 4manifolds
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ABSTRACT: The authors deduce two fundamental results which clarify from a new viewpoint that 4dimensional manifolds differ essentially from manifolds in any other dimension. Recall first that every pseudogroup of homeomorphisms of Euclidean space defines the corresponding category of manifolds. A homeomorphism ϕ with domain D⊂ℝ n is called quasiconformal if for all x in D lim sup r→0 max{ϕ(y)ϕ(x)yx=r} min{ϕ(y)ϕ(x)yx=r}≤K with some K≥1. Hence the category of quasiconformal manifolds is intermediate between the topological manifolds and the smooth manifolds. The second author deduced [Geometric topology, Proc. Conf., Athens/Ga. 1977, 543555 (1979; Zbl 0478.57007)] that for n≠4 any topological nmanifold admits a quasiconformal structure. Moreover, any two quasiconformal structures are equivalent by a homeomorphism isotopic to the identity. But for 4dimensional manifolds the following two results are proved in the present paper. I. There are topological 4manifolds which do not admit any quasiconformal structure. II. There are quasiconformal (indeed smooth) 4manifolds which are homeomorphic but not quasiconformally equivalent. The proofs are presented in detail.Acta Mathematica 01/1989; 163(1):181252. · 2.71 Impact Factor 
Article: Definitions of quasiconformality
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ABSTRACT: We establish that the infinitesimal “ H definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even in R n where we obtain that the “limsup” condition in the H definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chordarc surfaces is also given. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/46582/1/222_2005_Article_BF01241122.pdfInventiones mathematicae 12/1995; · 2.26 Impact Factor
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