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The intrinsic volume and the intrinsic Laplacian in sub-Riemannian geometry

Mercoledi' 16 marzo 2011 - Ugo Boscain

ARGOMENTI: Seminari

SEMINARI DI EQUAZIONI DIFFERENZIALI E APPLICAZIONI
Mercoledi' 16 marzo alle ore 14:30 in aula 2BC30 Ugo Boscain (Ecole Polytechnique, Palaiseau) terra' un seminario dal titolo "The intrinsic volume and the intrinsic Laplacian in sub-Riemannian geometry".

-Abstract
For a sub-Riemannian manifold (i.e. for a minimum time problem for a control system which is linear in the controls and with controls bounded in a disk) we study the problem of defining intrinsecally a way of measuring volumes. The main purpose is the one of defining intrinsically the sub-elliptic Laplacian. We compare two different notions of volumes namely the Popp volume and the Hausdorff volume. We prove that up to dimension 4 they are proportional. Starting from dimension 5, this is not true in general and moreover they are not smooth one to respect to the other. For the step 2 corank 1 case, we prove that the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to the Popp one is C3 but not C5 in general. This answer to a question formulated in the book of R. Montgomery.

Rif. int. M. Bardi, C. Marchi, G. Colombo

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