0000000000123750
AUTHOR
Pierre Pansu
Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds
Abstract We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumptions, this can happen only on topological spheres. To cite this article: R. Grimaldi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).
Bounded geometry, growth and topology
We characterize functions which are growth types of Riemannian manifolds of bounded geometry.
Semianalyticity of isoperimetric profiles
It is shown that, in dimensions $<8$, isoperimetric profiles of compact real analytic Riemannian manifolds are semi-analytic.
Sur la r�gularit� de la fonction croissance d'une vari�t� riemannienne
On etudie la differentiabilite de la fonction croissance d'une variete riemannienne complete. En general, elle a la meme regularite qu'une fonction concave: la derivee peut avoir des sauts pour lesquels on donne une formule. Dans le cas analytique reel, la fonction croissance est de classeC1. Un exemple montre qu'elle n'est pas necessairementC2. A titre d'application, nous construisons, pour toute variete ouverte paracompacteM et toute fonction croissantev de classeC1, une metrique continue de croissance egale av et une metrique de classeC∞ surM de croissance proche dev en topologieC1-fine.
Sard property for the endpoint map on some Carnot groups
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizat…