Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups

Details Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups

2010-12-11

arxiv.org/abs/1012.2442v2

Mathematics

Differential Geometry

43 pages

Scientific paper

Let G be a k-step Carnot group. We will prove an isoperimetric inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. The inequality is obtained by using the "natural" homogeneous measures on S and its boundary. This generalizes an inequality due to Michael and Simon, and Allard, independently. Some applications will be then discussed.

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