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Classes of Noncontact Mappings of Carnot Groups and Metric Properties

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Abstract

We study the metric properties of level surfaces for classes of smooth noncontact mappings from arbitrary Carnot groups into two-step ones with some constraints on the dimensions of horizontal subbundles and the subbundles corresponding to degree 2 fields. We calculate the Hausdorff dimension of the level surfaces with respect to the sub-Riemannian quasimetric and derive an analytical relation between the Hausdorff measures for the sub-Riemannian quasimetric and the Riemannian metric. As application, we establish a new form of coarea formula, also proving that the new coarea factor is well defined.

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References

  1. Karmanova M.B., “Sub-Riemannian properties of the level sets of noncontact mappings of Heisenberg groups,” Siberian Adv. Math., vol. 33, no. 2, 28–38 (2023).

    Article  MathSciNet  Google Scholar 

  2. Franchi B., Serapioni R., and Serra Cassano F., “Rectifiability and perimeter in the Heisenberg group,” Math. Ann., vol. 321, 479–531 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  3. Basalaev S.G., “One-dimensional level surfaces of \( hc \)-differentiable mappings on Carnot–Carathéodory spaces,” Vestn. NGU, vol. 13, no. 4, 16–36 (2013).

    MATH  Google Scholar 

  4. Kozhevnikov A., Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot. Géométrie métrique, Université Paris Sud, Paris (2015).

    Google Scholar 

  5. Franchi B. and Serapioni R., “Intrinsic Lipschitz graphs within Carnot groups,” J. Geom. Anal., vol. 26, no. 3, 1946–1994 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  6. Karmanova M.B., “On local metric characteristics of level sets of \( C^{1}_{H} \)-mappings of Carnot manifolds,” Sib. Math. J., vol. 60, no. 6, 1007–1021 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  7. Karmanova M.B., “Level sets of classes of mappings of two-step Carnot groups in a nonholonomic interpretation,” Sib. Math. J., vol. 60, no. 2, 304–311 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  8. Folland G.B. and Stein E.M., Hardy Spaces on Homogeneous Groups, Princeton Univ., Princeton (1982) (Princeton Math. Notes; vol. 28).

    MATH  Google Scholar 

  9. Pansu P., “Métriques de Carnot–Carathéodory et quasi-isométries des espaces symétriques de rang un,” Ann. Math., vol. 129, no. 1, 1–60 (1989) [French].

    Article  MathSciNet  MATH  Google Scholar 

  10. Karmanova M. and Vodopyanov S., “A coarea formula for smooth contact mappings of Carnot–Carathéodory spaces,” Acta Appl. Math., vol. 128, no. 1, 67–111 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  11. Vodopyanov S., “Geometry of Carnot–Carathéodory spaces and differentiability of mappings,” in: The Interaction of Analysis and Geometry. Contemporary Mathematics, vol. 424, Amer. Math. Soc., Providence (2007), 247–301 (Contemporary Mathematics; vol. 424).

  12. Karmanova M.B., “Sub-Lorentzian coarea formula for mappings of Carnot groups,” Sib. Math. J., vol. 63, no. 3, 485–508 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  13. Karmanova M.B., “The coarea formula for vector functions on Carnot groups with sub-Lorentzian structure,” Sib. Math. J., vol. 62, no. 2, 239–261 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  14. Vodopyanov S.K. and Ukhlov A.D., “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I,” Siberian Adv. Math., vol. 14, no. 4, 78–125 (2004).

    MathSciNet  MATH  Google Scholar 

  15. Vodopyanov S.K. and Ukhlov A.D., “Set functions and their applications in the theory of Lebesgue and Sobolev spaces. II,” Siberian Adv. Math., vol. 15, no. 1, 91–125 (2005).

    MathSciNet  Google Scholar 

  16. Federer H., Geometric Measure Theory, Springer, Berlin, Heidelberg, New York (1969).

    MATH  Google Scholar 

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Funding

The author was supported by the Mathematical Center in Akademgorodok under Agreement 075–15–2022–281 on 05.04.2022 with the Ministry of Science and Higher Education of the Russian Federation.

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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.

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As author of this work, I declare that I have no conflicts of interest.

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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1199–1223. https://doi.org/10.33048/smzh.2023.64.608

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Karmanova, M.B. Classes of Noncontact Mappings of Carnot Groups and Metric Properties. Sib Math J 64, 1330–1350 (2023). https://doi.org/10.1134/S0037446623060083

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  • DOI: https://doi.org/10.1134/S0037446623060083

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