On the Role of Riemannian Metrics in Conformal and Quasiconformal Geometry

Source: arXiv

ABSTRACT This article is the introductory part of authors PhD thesis. The article
presents a new coordinate invariant definition of quasiregular and
quasiconformal mappings on Riemannian manifolds that generalizes the definition
of quasiregular mappings on $\R^n$. The new definition arises naturally from
the inner product structures of Riemannian manifolds. The basic properties of
the mappings satisfying the new definition and a natural convergence theorem
for these mappings are given. These results are applied in a subsequent paper,
arXiv:1209.1285. In the current article, an application, likewise demonstrating
the usability of the new definition, is given. It is proven that any countable
quasiconformal group on a general Riemannian manifolds admits an invariant
conformal structure. This result generalizes a classical result by Pekka Tukia
in the countable case.

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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. 1960-1990, Lect. Notes Math. 1508, 93–103 (1992; Zbl 0764.30017)] of Rickman. The Rickman-Picard 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 n-manifold, n≥2· If there exists a nonconstant K-quasiregular 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):219-238. · 2.71 Impact Factor
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    ABSTRACT: The authors deduce two fundamental results which clarify from a new viewpoint that 4-dimensional 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)|||y-x|=r} min{|ϕ(y)-ϕ(x)|||y-x|=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, 543-555 (1979; Zbl 0478.57007)] that for n≠4 any topological n-manifold admits a quasiconformal structure. Moreover, any two quasiconformal structures are equivalent by a homeomorphism isotopic to the identity. But for 4-dimensional manifolds the following two results are proved in the present paper. I. There are topological 4-manifolds which do not admit any quasiconformal structure. II. There are quasiconformal (indeed smooth) 4-manifolds which are homeomorphic but not quasiconformally equivalent. The proofs are presented in detail.
    Acta Mathematica 01/1989; 163(1):181-252. · 2.71 Impact Factor
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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 chord-arc surfaces is also given. Peer Reviewed
    Inventiones mathematicae 12/1995; · 2.26 Impact Factor


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