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

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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

Full-text PDF (412 kB) Citations (29)

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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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Linking options:

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https://www.mathnet.ru/eng/smj/v53/i4/p839

This publication is cited in the following 29 articles:

M. B. Karmanova, “O minimalnykh poverkhnostyakh nad mnogoobraziyami Karno proizvolnoi glubiny”, Matem. tr., 25:1 (2022), 74–101
M. B. Karmanova, “Minimal Surfaces Over Carnot Manifolds”, Sib. Adv. Math., 32:3 (2022), 211
M. B. Karmanova, “Properties of minimal surfaces over depth 2 Carnot manifolds”, Siberian Math. J., 62:6 (2021), 1050–1062
M. B. Karmanova, “Two-step sub-Lorentzian structures and graph surfaces”, Izv. Math., 84:1 (2020), 52–94
M. B. Karmanova, “The area of graphs on arbitrary carnot groups with sub-lorentzian structure”, Siberian Math. J., 61:4 (2020), 648–670
M. B. Karmanova, “Classes of maximal surfaces on carnot groups”, Siberian Math. J., 61:5 (2020), 803–817
Maria Karmanova, Trends in Mathematics, Geometric Methods in Physics XXXVIII, 2020, 129
M. B. Karmanova, “Minimal graph-surfaces on arbitrary two-step Carnot groups”, Russian Math. (Iz. VUZ), 63:5 (2019), 13–26
M. B. Karmanova, “On the class of Hölder surfaces in Carnot–Carathéodory spaces”, Siberian Math. J., 60:5 (2019), 861–885
M. B. Karmanova, “Area of graph surfaces on Carnot groups with sub-Lorentzian structure”, 99, no. 2, 2019, 145–148
M. B. Karmanova, “Maximal surfaces on five-dimensional group structures”, Siberian Math. J., 59:3 (2018), 442–457
M. B. Karmanova, “Three-dimensional graph surfaces on five-dimensional Carnot–Carathéodory spaces”, Siberian Math. J., 59:4 (2018), 657–676
M. B. Karmanova, “Polynomial sub-Riemannian differentiability on Carnot–Carathéodory spaces”, Siberian Math. J., 59:5 (2018), 860–869
M. B. Karmanova, “Hölder mappings of Carnot groups and intrinsic bases”, Dokl. Math., 95:1 (2017), 1–4
M. B. Karmanova, “Approximation of Hölder mappings on Carnot–Carathéodory spaces”, Dokl. Math., 95:3 (2017), 199–202
M. B. Karmanova, “Metric properties of classes of Hölder surfaces on Carnot groups”, Dokl. Math., 95:2 (2017), 118–121
M. B. Karmanova, “Graph surfaces on five-dimensional sub-Lorentzian structures”, Siberian Math. J., 58:1 (2017), 91–108
M. B. Karmanova, “The polynomial sub-Riemannian differentiability of some Hölder mappings of Carnot groups”, Siberian Math. J., 58:2 (2017), 232–254
M. B. Karmanova, “Area formulas for classes of Hölder continuous mappings of Carnot groups”, Siberian Math. J., 58:5 (2017), 817–836
M. B. Karmanova, “Maximal graph surfaces on four-dimensional two-step sub-Lorentzian structures”, Siberian Math. J., 57:2 (2016), 274–284

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

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