On Rectifiable Measures in Carnot Groups: Existence of Density

Area of Science:

• Geometric Measure Theory
• Analysis on Carnot Groups
• Non-Euclidean Geometry

Background:

• Existing notions of rectifiability in Carnot groups lack a comprehensive framework.
• Understanding measure properties in non-Euclidean settings is crucial for geometric analysis.

Purpose of the Study:

• Introduce and define a new concept of h-rectifiability for Radon measures in Carnot groups.
• Compare this new definition with existing rectifiability notions.
• Investigate the structural properties of h-rectifiable measures.

Main Methods:

• Definition of h-rectifiability based on h-lower and h-upper densities.
• Analysis of tangent measures at almost every point.
• Comparison with previously established rectifiability concepts.
• Characterization of the support of h-rectifiable measures.

Main Results:

• The proposed h-rectifiability is strictly weaker than existing notions in Carnot groups.
• The support of an h-rectifiable measure is covered by sets with a cone-like property.
• h-rectifiable measures with complemented tangents are supported on Lipschitz and differentiable graphs.

Conclusions:

• The new definition of h-rectifiability provides a more general framework.
• The structural properties reveal insights into the geometry of these measures.
• The findings contribute to the understanding of measure theory in Carnot groups.