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The graphs of Lipschitz functions and minimal surfaces on Carnot groups

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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).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    M. B. Karmanova

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  1. M. B. Karmanova
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Correspondence to M. B. Karmanova.

Additional information

Original Russian Text Copyright © 2012 Karmanova M.B.

The author was supported by the State Maintenance Program for the Junior Scientists and the Leading Scientific Schools of the Russian Federation (Grant NSh-921.2012.1) and the Integration Grant of the Siberian and Far East Divisions of the Russian Academy of Sciences (No. 56).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 4, pp. 839–861, July–August, 2012.

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Karmanova, M.B. The graphs of Lipschitz functions and minimal surfaces on Carnot groups. Sib Math J 53, 672–690 (2012). https://doi.org/10.1134/S0037446612040106

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  • DOI: https://doi.org/10.1134/S0037446612040106

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