Submitted Paper
Inserted: 23 apr 2003
Year: 2003
Abstract:
We study the relationship between the geometry of $C^2$ hypersurfaces in a Carnot-Carathéodory space and the Ahlfors regularity of the corresponding perimeter measure. In addition, we find a sharp geometric condition on the nature of the characteristic set which implies the $1$-Ahlfors regularity. As a corollary, we have that all $C^2$ hypersurfaces in a Carnot-Carathéodory space of rank two (or less) are $1-$Ahlfors regular. In particular, in the notation used by David and Semmes \cite{DS2}, in a Carnot group of step $2$ with homogeneous dimension $Q$, all $C^{1,1}$ hypersurfaces are (Ahlfors) regular of dimension $Q-1$.
Download: