A blow-up theorem for regular hypersurfaces on nilpotent groups
``The project to develop classical tools of geometric measure theory in stratified Lie groups, also known as Carnot groups and in the more general Carnot-Carathéodory spaces, is at an embryonic stage'', says the author. He begins his article by finding the geometrical meaning of a certain integral which appears in isoperimetric inequalities formulated in stratified groups. He reformulates the integral in terms of a left invariant Riemannian metric (precisely graded metrics) and a homogeneous distance of the group. He introduces a ``metric factor'' into his formula to take into account both the anisotropy of the homogeneous distance and the position of the tangent space of a hypersurface of class \(C^1\) in \(\mathbb{R}^n\). The main tool is the blow-up theorem (2.1) on the Riemannian surface measure with respect to a fixed homogeneous distance. In subsection 2, the author introduces \(R\)-invariant distances and \(R\)-rotational groups with the property of having a metric factor which is a dimensional constant independent of the direction and he also presents important examples where these properties hold. As a variant of the blow-up theorem he obtains (3.1) that the characteristic set of a \(C^{1,1}\) hypersurface in a two-step graded group has finite \(q-2\) Hausdorff measure and its Hausdorff dimension does not exceed \(q-2\) (this estimate is also optimal). By virtue of the blow-up theorem he also derives a version of the coarea formula for a Lipschitz map on stratified groups.
scientific article; zbMATH DE number 1889226