## Isoperimetric, Sobolev and Poincar\'e inequalities on hypersurfaces in sub-Riemannian Carnot groups

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Montefalcone, F.

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In this paper we shall study smooth submanifolds immersed in a k-step Carnot
group G of homogeneous dimension Q. Among other results, we shall prove an
isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S
with - or without - boundary $\partial S$; S and $\partial S$ are endowed with
their homogeneous measures, actually equivalent to the intrinsic
(Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some
homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical
inequality, involving the mean curvature of the hypersurface, proven by Michael
and Simon [63] and, independently by Allard [1]. In particular, from this
result one may deduce some related Sobolev-type inequalities; see Section 7.
The strategy of the proof is inspired by the classical one. In particular, we
shall begin by proving some linear isoperimetric inequalities. Once this is
proven, one can deduce a local monotonicity formula and then conclude the proof
by a covering argument. We stress however that there are many differences, due
to our different geometric setting. Some of the tools which have been developed
ad hoc in this paper are, in order, a ``blow-up'' theorem, which also holds for
characteristic points, and a smooth Coarea Formula for the HS-gradient; see
Section 3 and Section 4. Other tools are the horizontal integration by parts
formula and the 1st variation of the H-perimeter already developed in [68],
[69], and here generalized to hypersurfaces having non-empty characteristic
set. Some natural applications of these results are in the study of minimal and
constant horizontal mean curvature hypersurfaces. Moreover we shall prove some
purely horizontal, local and global Poincar\'e-type inequalities as well as
some related facts and consequences; see Section 4 and Section 5.

Comment: 61 pages

Comment: 61 pages

##### Keywords

Mathematics - Analysis of PDEs, Mathematics - Differential Geometry, 49Q15, 46E35, 22E60