Blow-up estimates at horizontal points and applications

TL;DR

This paper investigates blow-up estimates at horizontal points of smooth submanifolds in stratified groups, providing new pointwise measure estimates and applications to intrinsic geometry and measure computations.

Contribution

It introduces pointwise estimates for Riemannian surface measure at horizontal points in two-step groups and establishes an integral formula for spherical Hausdorff measure of smooth submanifolds.

Findings

Pointwise measure estimates at horizontal points in two-step groups

An integral formula for spherical Hausdorff measure of submanifolds

Existence of intrinsic blow-ups for $C^2$ smooth submanifolds

Abstract

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of $C^{1,1}$ smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any $C^{1,1}$ submanifold. Our technique also shows that $C^2$ smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety.