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Sibirsk. Mat. Zh., 2012, Volume 53, Number 4, Pages 839–861 (Mi smj2368)  

This article is cited in 19 scientific papers (total in 19 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

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.

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English version:
Siberian Mathematical Journal, 2012, 53:4, 672–690

Bibliographic databases:

UDC: 517.518.1+514.76
Received: 12.03.2012

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
\Bibitem{Kar12}
\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://www.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}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84865504640}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. B. Karmanova, “An area formula for Lipschitz mappings of Carnot–Carathéodory spaces”, Izv. Math., 78:3 (2014), 475–499  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. M. B. Karmanova, “The area formula for graphs on $4$-dimensional $2$-step sub-Lorentzian structures”, Siberian Math. J., 56:5 (2015), 852–871  mathnet  crossref  crossref  isi  elib  elib
    3. M. B. Karmanova, “Graph surfaces over three-dimensional Lie groups with sub-Riemannian structure”, Siberian Math. J., 56:6 (2015), 1080–1092  mathnet  crossref  crossref  mathscinet  isi  elib
    4. M. B. Karmanova, “Graph surfaces over three-dimensional Carnot–Carathéodory spaces”, Dokl. Math., 92:1 (2015), 439–442  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    5. M. B. Karmanova, “The area of graph surfaces on four-dimensional two-step sub-Lorentzian structures”, Dokl. Math., 92:1 (2015), 456–459  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. 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
    7. M. B. Karmanova, “Polynomial sub-Riemannian differentiability on Carnot groups”, Dokl. Math., 94:3 (2016), 663–666  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    8. M. B. Karmanova, “Variations of nonholonomic-valued mappings and their applications to maximal surface theory”, Dokl. Math., 93:3 (2016), 276–279  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    9. M. B. Karmanova, “Graph surfaces of codimension two over three-dimensional Carnot–Carathéodory spaces”, Dokl. Math., 93:3 (2016), 322–325  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    10. M. B. Karmanova, “Area formula for graph surfaces on five-dimensional sub-Lorentzian structures”, Dokl. Math., 93:2 (2016), 216–219  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    11. 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
    12. 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
    13. 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
    14. 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
    15. 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
    16. 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
    17. M. B. Karmanova, “Maximal surfaces on five-dimensional group structures”, Siberian Math. J., 59:3 (2018), 442–457  mathnet  crossref  crossref  isi  elib
    18. 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
    19. M. B. Karmanova, “Polynomial sub-Riemannian differentiability on Carnot–Carathéodory spaces”, Siberian Math. J., 59:5 (2018), 860–869  mathnet  crossref  crossref  isi
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