Surface measures in Carnot-Carath�odory spaces

Abstract

In the framework of Carnot-Carathéodory spaces we study Minkowski content and perimeter, we prove some coarea formulas, and finally we prove some variational approximations of the perimeter.

Geodesics in the Heisenberg Group

“…This improves a result of Monti [12,13] who proved that this distance is C ∞ smooth away from the center. We also nd a formula for the Carnot-Carathéodory distance (Corollary 3.2) that, we hope, will nd application in the study of geometric properties of the Heisenberg groups.…”

“…It is well known that the distance function in H n is C ∞ smooth away from the center [12,13], but through the use of (2.2), we will now see that this distance function is actually real analytic.…”

“…For the sake of completeness, we decided to provide a proof of this result in the Appendix. For a proof of (1) in more generality, see Theorem 1.3.5 in [12].…”

On Conditions for Unrectifiability of a Metric Space

“…Moreover H (γ) equals the length cc (γ) of γ with respect to the CarnotCarathéodory metric. A detailed account on this topic can be found in [20]. id : (Ω, dcc) → Ω from the CarnotCarathéodory space onto Ω with Euclidean metric is locally weakly BLD.…”

Regularity of the distance function to smooth hypersurfaces in some two-step Carnot groups

“…Let G be a 2-step Carnot group. The calculation below is mainly based on the eikonal equation, first proven in [20]. Concerning the case of the Heisenberg group see also [19].…”