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The graphs of Lipschitz functions and minimal surfaces on Carnot groups. (English. Russian original) Zbl 1262.53029

Sib. Math. J. 53, No. 4, 672-690 (2012); translation from Sib. Mat. Zh. 53, No. 4, 839-861 (2012).
Given a Carnot group \({\mathbb G}\), a nilpotent graded Lie group \(\tilde{\mathbb G}\), \(E\subset {\mathbb G}\), and a map \(\varphi:E\to \tilde{\mathbb G}\), the author introduces the notion of “polynomially \(hc\)-differentiability” of \(\varphi\) with respect to a quasimetric on \(\varphi(E)\times\varphi(E)\) in Definition 9. Lipschitz mappings with respect to the sub-Riemannian structures satisfy this property. Under the technical assumptions given in Description 1 of the paper, the author proves in Theorem 2 the polynomially \(hc\)-differentiability of \(\varphi_\Gamma:D\subset {\mathbb G}\to\tilde{\mathbb G}\) (Definition 10) and an area formula for the intrinsic measure (Definition 12). Variational formulas for the intrinsic measure and consequences are obtained in Theorem 3.
Related variational formulas for graphs in the sub-Riemannian Heisenberg group can be found by J.-H. Cheng et al. [Math. Ann. 337, No. 2, 253–293 (2007; Zbl 1109.35009)] and R. Monti et al. [Boll. Unione Mat. Ital. (9) 1, No. 3, 709–727 (2008; Zbl 1204.53048)].

MSC:

53C17 Sub-Riemannian geometry
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q05 Minimal surfaces and optimization
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
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