A new class of \((\mathcal{H}^k,1)\)-rectifiable subsets of metric spaces

Roberta Ghezzi


Event: ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

Date: Oct 11, 2011, time: 17:10

Place: Centro De Giorgi, Scuola Normale Superiore

Abstract. The main motivation of this talk arises from the study of Carnot-Carathéodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed.
This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree \(k\), which are Hölder but not Lipschitz continuous when \(k>1\). Replacing Lipschitz curves by this kind of curves we define \((\mathcal{H}^k,1)\)-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry.
This theorem is a consequence of computations of Hausdorff measures along
curves, for which we give an integral formula.
In particular, we show that both spherical and usual Hausdorff
measures along curves coincide with a class of dimensioned lengths and are
related to an interpolation complexity, for which estimates have already
been obtained in Carnot-Carathéodory spaces.