On minimal surfaces on two-step Carnot groups

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Abstract


For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of minimal surfaces’ problem are found. A suitable notion of the (sub-Riemannian) area functional increment is introduced, differentiability of this functional is proved, and necessary minimality conditions are deduced. They are also expressed in terms of sub-Riemaninan mean curvature.

 


About the authors

M. B. Karmanova

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

Author for correspondence.
Email: maryka@math.nsc.ru

Russian Federation, 4, Acad. Koptyug prospect, Novosibirsk, 630090

References

  1. Karmanova M., Vodopyanov S. Geometry of Carnot-Carathéodory Spaces, Differentiability, Coarea and Area Formulas. In: Analysis and Mathematical Phy-sics. Basel: Birkhäuser, 2009. P. 233-335.
  2. Карманова М. Б. Минимальные поверхности-графики на произвольных двуступенчатых группах Карно // Изв. вузов. Математика. 2019.
  3. Folland G.B., Stein E. M. Hardy Spaces on Homogeneous Groups. Princeton: Princeton Univ. Press, 1982.
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  5. Vodopyanov S. // Contemp. Math. 2007. V. 424. P. 247-301.
  6. Карманова М.Б. // Сиб. мат. журн. 2018. Т. 59. № 4. С. 834-857.
  7. Карманова М.Б. // ДАН. 2018. Т. 480. № 1. С. 16-20.

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