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On rectifiable measures in Carnot groups: representation.

Gioacchino Antonelli1, Andrea Merlo2

  • 1Scuola Normale Superiore, Piazza dei Cavalieri, 7, Pisa, 56126 Italy.

Calculus of Variations and Partial Differential Equations
|December 6, 2021
PubMed
Summary
This summary is machine-generated.

This study establishes the equivalence of infinitesimal and global definitions for rectifiable measures in Carnot groups. It also proves a geometric area formula, showing geodesic spheres are intrinsically rectifiable.

Keywords:
22E2526A1628A7549Q1553C17

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Area of Science:

  • Differential Geometry
  • Geometric Measure Theory
  • Harmonic Analysis

Background:

  • The study of rectifiability is crucial in geometric measure theory, particularly in understanding lower-dimensional structures within higher-dimensional spaces.
  • Carnot groups are a fundamental class of non-Riemannian manifolds with applications in various fields, including analysis and differential geometry.
  • Existing theories on rectifiability in Carnot groups often rely on specific definitions and may not cover all cases.

Purpose of the Study:

  • To establish the equivalence between infinitesimal and global definitions of rectifiable measures in arbitrary Carnot groups.
  • To derive a geometric area formula for intrinsically differentiable graphs in Carnot groups.
  • To prove the intrinsic rectifiability of preimages of Lipschitz functions and geodesic spheres in Carnot groups.

Main Methods:

  • Investigating the relationship between flat tangent measures and coverings by intrinsically differentiable graphs.
  • Developing and applying a geometric area formula for Hausdorff measure on intrinsically differentiable graphs.
  • Utilizing techniques from geometric measure theory and the theory of Carnot groups.

Main Results:

  • Demonstrated the equivalence of infinitesimal and global definitions of rectifiable measures in Carnot groups, using intrinsically differentiable graphs.
  • Established a geometric area formula for centered Hausdorff measure on intrinsically differentiable graphs, extending prior results.
  • Proved the intrinsic rectifiability of almost all preimages of Lipschitz functions between Carnot groups.
  • Showed that geodesic spheres in Carnot groups equipped with homogeneous distances are intrinsically rectifiable.

Conclusions:

  • The equivalence of rectifiability definitions provides a more robust theoretical framework for Carnot groups.
  • The new area formula offers a powerful tool for analyzing geometric structures within these groups.
  • The intrinsic rectifiability results have significant implications for understanding the geometry of sets and functions in Carnot groups, particularly for geodesic spheres.