Sub-Riemannian calculus on hypersurfaces in Carnot groups

By D. Danielli, Nicola Garofalo, D. Nhieu. 2006

137 citations recorded by the source corpus.

We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace–Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/fa84428e8572bf576684146871fb56cff068bb3d

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Citations in the corpus

Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity (2008, methodology): “we can define the horizontal unit normal vector νH, as in [12], [30] and [23] by”
Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups (2017, background): “Gromov [28] or Danielli, Garofalo and Nhieu [9] for general expositions from different points of view), but also in application settings such as mathematical models of crystal material and human vision (see, for example, [6] and [7]).”
Area-stationary surfaces in the Heisenberg group H1 (2005, methodology): “Finally, by using the description of the singular set in [CHMY], the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify volumepreserving area-stationary surfaces in H1 with non-empty singular set.”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “To quote a few: H0 can be defined in terms of the first variation of the area functional [20, 29, 9, 36, 40, 31, 7], as horizontal divergence of the horizontal unit normal or as limit of the mean curvatures Hε in the Riemannian metrics gε as ε → 0.”
Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, background): “We want to mention that the expression we got for the intrinsic Gaussian curvature K0 appears also in the paper [19] in the study of stability properties of minimal surfaces in H, and also in the upcoming manuscript [16].”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, background): “Several first variation formula have been independently established in receent years, see for instance [18], [7], [8], [4], [32], [33], [27], [28], [35], [36].”