Slow mappings of finite distortion

Onninen, Jani; Pankka, Pekka

doi:10.1007/s00208-011-0751-3

Cited by 3 publications

(2 citation statements)

References 22 publications

(22 reference statements)

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“…Rational homology spheres were investigated in [8,10,12] in the context of the degree theory of Orlicz-Sobolev mappings. Quasiregular mappings and mappings of finite distortion with values into rational homology spheres have also been studied in [5,29]. For more information about rational homology spheres, see Section 2.…”

Section: Introductionmentioning

confidence: 99%

Topological obstructions to continuity of Orlicz–Sobolev mappings of finite distortion

Goldstein

Hajłasz

2018

Annali di Matematica

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In the paper we investigate continuity of Orlicz-Sobolev mappings W 1,P (M, N ) of finite distortion between smooth Riemannian n-manifolds, n ≥ 2, under the assumption that the Young function P satisfies the so called divergence condition ∞ 1 P (t)/t n+1 dt = ∞. We prove that if the manifolds are oriented, N is compact, and the universal cover of N is not a rational homology sphere, then such mappings are continuous. That includes mappings with Df ∈ L n and, more generally, mappings with Df ∈ L n log −1 L. On the other hand, if the space W 1,P is larger than W 1,n (for example if Df ∈ L n log −1 L), and the universal cover of N is homeomorphic to S n , n = 4, or is diffeomorphic to S n , n = 4, then we construct an example of a mapping in W 1,P (M, N ) that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold N .Recently, the result of Vodop'janov and Gol'dšteȋn has been extended to the case of mappings between manifolds [9].Theorem 1. Let M and N be smooth, oriented, n-dimensional Riemannian manifolds without boundary and assume additionally that N is compact. If f ∈ W 1,n (M, N) has 2010 Mathematics Subject Classification. Primary 30C65; Secondary 46E35, 58C07.

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Section: Introductionmentioning

confidence: 99%

Topological obstructions to continuity of Orlicz–Sobolev mappings of finite distortion

Goldstein

Hajłasz

2018

Annali di Matematica

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“…The class of quasiregular mappings and mappings of finite distortion between manifolds have been studied by many authors, see for example [2,9,10,14,15] and references therein. In most of the papers the target manifold is compact.…”

Section: Introductionmentioning

confidence: 99%

Finite Distortion Sobolev Mappings between Manifolds are Continuous

Goldstein

Hajłasz

Pakzad

2017

International Mathematics Research Notices

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Abstract. We prove that if M and N are Riemannian, oriented n-dimensional manifolds without boundary and additionally N is compact, then Sobolev mappings W 1,n (M, N ) of finite distortion are continuous. In particular, W 1,n (M, N ) mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings W 1,n (Ω, R n ), where Ω ⊂ R n is an open set. The case of mappings between manifolds is much more difficult.

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Signed quasiregular curves

Heikkilä¹

2023

JAMA

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Slow mappings of finite distortion

Abstract: Abstract. We examine mappings of finite distortion between Riemannian manifolds. We use integral type isoperimetric inequalities to obtain Liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds.

Cited by 3 publications

References 22 publications

Topological obstructions to continuity of Orlicz–Sobolev mappings of finite distortion

Topological obstructions to continuity of Orlicz–Sobolev mappings of finite distortion

Finite Distortion Sobolev Mappings between Manifolds are Continuous

Signed quasiregular curves

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