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Sibirskii Matematicheskii Zhurnal, 2012, Volume 53, Number 4, Pages 839–861 (Mi smj2368)  

This article is cited in 29 scientific papers (total in 29 papers)

The graphs of Lipschitz functions and minimal surfaces on Carnot groups

M. B. Karmanova

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
References:
Abstract: We study and solve a new problem for the class of Lipschitz mappings (with respect to sub-Riemannian metrics) on Carnot groups. We introduce the new concept of graph for the functions on a Carnot group, and then the new concept of sub-Riemannian differentiability generalizing hc-differentiability. We prove that the mapping-“graphs” are almost everywhere differentiable in the new sense. For these mappings we define a concept of intrinsic measure and obtain an area formula for calculating this measure. By way of application, we find necessary and sufficient conditions on the class of surface-“graphs” under which they are minimal surfaces (with respect to the intrinsic measure of a surface).
Keywords: Carnot group, Lipschitz mapping, graph, area formula, minimal surface.
Received: 12.03.2012
English version:
Siberian Mathematical Journal, 2012, Volume 53, Issue 4, Pages 672–690
DOI: https://doi.org/10.1134/S0037446612040106
Bibliographic databases:
Document Type: Article
UDC: 517.518.1+514.76
Language: Russian
Citation: M. B. Karmanova, “The graphs of Lipschitz functions and minimal surfaces on Carnot groups”, Sibirsk. Mat. Zh., 53:4 (2012), 839–861; Siberian Math. J., 53:4 (2012), 672–690
Citation in format AMSBIB
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\by M.~B.~Karmanova
\paper The graphs of Lipschitz functions and minimal surfaces on Carnot groups
\jour Sibirsk. Mat. Zh.
\yr 2012
\vol 53
\issue 4
\pages 839--861
\mathnet{http://mi.mathnet.ru/smj2368}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3013531}
\transl
\jour Siberian Math. J.
\yr 2012
\vol 53
\issue 4
\pages 672--690
\crossref{https://doi.org/10.1134/S0037446612040106}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000307983400010}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84865504640}
Linking options:
  • https://www.mathnet.ru/eng/smj2368
  • https://www.mathnet.ru/eng/smj/v53/i4/p839
  • This publication is cited in the following 29 articles:
    1. M. B. Karmanova, “O minimalnykh poverkhnostyakh nad mnogoobraziyami Karno proizvolnoi glubiny”, Matem. tr., 25:1 (2022), 74–101  mathnet  crossref
    2. M. B. Karmanova, “Minimal Surfaces Over Carnot Manifolds”, Sib. Adv. Math., 32:3 (2022), 211  crossref
    3. M. B. Karmanova, “Properties of minimal surfaces over depth 2 Carnot manifolds”, Siberian Math. J., 62:6 (2021), 1050–1062  mathnet  crossref  crossref  isi  elib
    4. M. B. Karmanova, “Two-step sub-Lorentzian structures and graph surfaces”, Izv. Math., 84:1 (2020), 52–94  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. M. B. Karmanova, “The area of graphs on arbitrary carnot groups with sub-lorentzian structure”, Siberian Math. J., 61:4 (2020), 648–670  mathnet  crossref  crossref  isi  elib
    6. M. B. Karmanova, “Classes of maximal surfaces on carnot groups”, Siberian Math. J., 61:5 (2020), 803–817  mathnet  crossref  crossref  isi
    7. Maria Karmanova, Trends in Mathematics, Geometric Methods in Physics XXXVIII, 2020, 129  crossref
    8. M. B. Karmanova, “Minimal graph-surfaces on arbitrary two-step Carnot groups”, Russian Math. (Iz. VUZ), 63:5 (2019), 13–26  mathnet  crossref  crossref  isi
    9. M. B. Karmanova, “On the class of Hölder surfaces in Carnot–Carathéodory spaces”, Siberian Math. J., 60:5 (2019), 861–885  mathnet  crossref  crossref  isi  elib
    10. M. B. Karmanova, “Area of graph surfaces on Carnot groups with sub-Lorentzian structure”, 99, no. 2, 2019, 145–148  crossref  crossref  zmath  isi  elib  scopus
    11. M. B. Karmanova, “Maximal surfaces on five-dimensional group structures”, Siberian Math. J., 59:3 (2018), 442–457  mathnet  crossref  crossref  isi  elib
    12. M. B. Karmanova, “Three-dimensional graph surfaces on five-dimensional Carnot–Carathéodory spaces”, Siberian Math. J., 59:4 (2018), 657–676  mathnet  crossref  crossref  isi  elib
    13. M. B. Karmanova, “Polynomial sub-Riemannian differentiability on Carnot–Carathéodory spaces”, Siberian Math. J., 59:5 (2018), 860–869  mathnet  crossref  crossref  isi  elib
    14. M. B. Karmanova, “Hölder mappings of Carnot groups and intrinsic bases”, Dokl. Math., 95:1 (2017), 1–4  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    15. M. B. Karmanova, “Approximation of Hölder mappings on Carnot–Carathéodory spaces”, Dokl. Math., 95:3 (2017), 199–202  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    16. M. B. Karmanova, “Metric properties of classes of Hölder surfaces on Carnot groups”, Dokl. Math., 95:2 (2017), 118–121  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    17. M. B. Karmanova, “Graph surfaces on five-dimensional sub-Lorentzian structures”, Siberian Math. J., 58:1 (2017), 91–108  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    18. M. B. Karmanova, “The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups”, Siberian Math. J., 58:2 (2017), 232–254  mathnet  crossref  crossref  isi  elib  elib
    19. M. B. Karmanova, “Area formulas for classes of Hölder continuous mappings of Carnot groups”, Siberian Math. J., 58:5 (2017), 817–836  mathnet  crossref  crossref  isi  elib  elib
    20. M. B. Karmanova, “Maximal graph surfaces on four-dimensional two-step sub-Lorentzian structures”, Siberian Math. J., 57:2 (2016), 274–284  mathnet  crossref  crossref  mathscinet  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
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