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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 5–32 (Mi izv2842)  

This article is cited in 10 scientific papers (total in 10 papers)

Spaces of differential forms and maps with controlled distortion

S. K. Vodop'yanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We study necessary and sufficient conditions for an approximately differentiable map $f\colon\mathbb M\to\mathbb M'$ between Riemannian manifolds to induce a bounded transfer operator of differential forms with respect to the norms of Lebesgue spaces. As a corollary, we see that every homeomorphism $f\colon\mathbb M\to\mathbb M'$ of class $\operatorname{ACL}(\mathbb M)$ whose transfer operator of differential forms with norm in $\mathcal L_p$ is an isomorphism must necessarily be either quasi-conformal or quasi-isometric. We give some applications of our results to the study of the functoriality of cohomology in Lebesgue spaces.

Keywords: Lebesgue space of differential forms, distortion of a map, quasi-conformal mapping, cohomology of Riemannian spaces.

DOI: https://doi.org/10.4213/im2842

Full text: PDF file (747 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:4, 663–689

Bibliographic databases:

UDC: 515.164.13+517.548.2
MSC: 46E35, 47B38, 30C65
Received: 27.06.2008

Citation: S. K. Vodop'yanov, “Spaces of differential forms and maps with controlled distortion”, Izv. RAN. Ser. Mat., 74:4 (2010), 5–32; Izv. Math., 74:4 (2010), 663–689

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vodopyanov S.K., “Mappings of finite codistortion and Sobolev classes of functions”, Dokl. Math, 84:2 (2011), 640–644  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    2. S. K. Vodopyanov, “Regularity of mappings inverse to Sobolev mappings”, Sb. Math., 203:10 (2012), 1383–1410  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Terenteva Yu.V., “Issledovanie svoistv integriruemosti proizvodnykh resheniya sopryazhënnogo (nelineinogo) uravneniya beltrami v sluchae vyrozhdeniya na granichnoi duge”, Vestnik rossiiskogo universiteta druzhby narodov. seriya: matematika, informatika, fizika, 2013, no. 1, 5–18  elib
    4. Vodopyanov S.K., “On admissible changes of variables for Sobolev functions on (sub)Riemannian manifolds”, Dokl. Math., 93:3 (2016), 318–321  crossref  mathscinet  zmath  isi  elib  scopus
    5. S. K. Vodop'yanov, N. A. Kudryavtseva, “On the Convergence of Mappings with $k$-Finite Distortion”, Math. Notes, 102:6 (2017), 878–883  mathnet  crossref  crossref  isi  elib
    6. N. A. Kudryavtseva, S. K. Vodopyanov, “On the convergence of mappings with $k$-finite distortion”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 88–100  mathnet  crossref  elib
    7. S. K. Vodopyanov, “Basics of the quasiconformal analysis of a two-index scale of spatial mappings”, Siberian Math. J., 59:5 (2018), 805–834  mathnet  crossref  crossref  isi  elib
    8. S. K. Vodopyanov, “Differentiability of mappings of the Sobolev space $W^1_{n-1}$ with conditions on the distortion function”, Siberian Math. J., 59:6 (2018), 983–1005  mathnet  crossref  crossref  isi  elib
    9. S. K. Vodopyanov, “Foundations of quasiconformal analysis of a two-index scale of spatial mappings”, Dokl. Math., 99:1 (2019), 23–27  crossref  crossref  zmath  isi  elib  scopus
    10. Molchanova A., Vodopyanov S., “Injectivity Almost Everywhere and Mappings With Finite Distortion in Nonlinear Elasticity”, Calc. Var. Partial Differ. Equ., 59:1 (2019), 17  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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