Montefalcone, F.
(2005)
*Alcune formule integrali nei Gruppi di Carnot.*
Seminario di Analisi Matematica- Dipartimento di Matematica dell' Università di Bologna- Anno Accademico 2004/05
.
pp. 1-21.

*This is the most updated version of the document.*

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## Abstract

Let G a k-step Carnot group. In the 1st part of this talk we are concerned about Integral
Geometry in the setting of Carnot groups. We start by illustrating an interplay between
volume and H-perimeter, using one-dimensional horizontal slicing. This result is a kind
of Fubini Theorem for H-regular hypersurfaces. Some applications are given: slicing of
HBV-functions, integral geometric formulae for volume and H-perimeter and, making use
of a suitable notion of convexity, we state a Cauchy type formula for this class of convex
sets. We then state a sub-Riemannian Santalò formula showing some related applications:
in particular we find two lower bounds for the 1st eigenvalue of the Dirichlet problem for
the Carnot sub-Laplacian on smooth domains. In the second part we introduce some
differential-geometric tools useful in the study of regular non-characteristic hypersurfaces.
In particular, we state divergence-type theorems and integration by parts formulas with
respect to the intrinsic H-perimeter measure
on hypersurfaces. Finally we shall give a general formula
for the 1st variation of the H-perimeter measure.

Abstract

Let G a k-step Carnot group. In the 1st part of this talk we are concerned about Integral
Geometry in the setting of Carnot groups. We start by illustrating an interplay between
volume and H-perimeter, using one-dimensional horizontal slicing. This result is a kind
of Fubini Theorem for H-regular hypersurfaces. Some applications are given: slicing of
HBV-functions, integral geometric formulae for volume and H-perimeter and, making use
of a suitable notion of convexity, we state a Cauchy type formula for this class of convex
sets. We then state a sub-Riemannian Santalò formula showing some related applications:
in particular we find two lower bounds for the 1st eigenvalue of the Dirichlet problem for
the Carnot sub-Laplacian on smooth domains. In the second part we introduce some
differential-geometric tools useful in the study of regular non-characteristic hypersurfaces.
In particular, we state divergence-type theorems and integration by parts formulas with
respect to the intrinsic H-perimeter measure
on hypersurfaces. Finally we shall give a general formula
for the 1st variation of the H-perimeter measure.

Document type

Article

Creators

Keywords

Carnot Groups, Integral Geometry, Sub-Riemannian Geometry, BV spaces, H-perimeter, 1st variation

Subjects

DOI

Deposit date

04 Feb 2008

Last modified

16 May 2011 12:07

URI

## Other metadata

Document type

Article

Creators

Keywords

Carnot Groups, Integral Geometry, Sub-Riemannian Geometry, BV spaces, H-perimeter, 1st variation

Subjects

DOI

Deposit date

04 Feb 2008

Last modified

16 May 2011 12:07

URI

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