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Horizontally Affine Functions on Step-2 Carnot Algebras.

Enrico Le Donne1,2, Daniele Morbidelli3, Séverine Rigot4

  • 1Département de Mathématiques, Université de Fribourg, Ch. du musée 23, 1700 Fribourg, Switzerland.

Journal of Geometric Analysis
|September 13, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces horizontally affine (h-affine) functions on step-2 Carnot algebras. We describe these functions and their connection to precisely monotone sets and minimal hypersurfaces.

Keywords:
Horizontally affine functionsStep-2 Carnot algebrasStep-2 Carnot groups

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Area of Science:

  • Mathematics
  • Geometric Analysis
  • Algebraic Geometry

Background:

  • Carnot algebras are fundamental in geometric analysis.
  • Horizontally affine functions (h-affine) have emerging applications.
  • Precisely monotone sets and minimal hypersurfaces are active research areas.

Purpose of the Study:

  • Introduce and characterize horizontally affine (h-affine) functions on step-2 Carnot algebras.
  • Establish an isomorphism between the vector space of h-affine functions and exterior algebras.
  • Describe h-affine functions on arbitrary step-2 Carnot algebras by leveraging results from free Carnot algebras.

Main Methods:

  • Definition and analysis of horizontally affine functions.
  • Isomorphism theorems relating function spaces to exterior algebras.
  • Deductive reasoning from free Carnot algebras to arbitrary ones.

Main Results:

  • A complete description of h-affine functions on step-2 Carnot algebras.
  • The vector space of h-affine functions on free step-2 rank-n Carnot algebras is isomorphic to the exterior algebra of .
  • Characterizations of step-2 Carnot algebras where h-affine functions are affine in the usual sense.

Conclusions:

  • H-affine functions provide a new perspective on Carnot algebra structures.
  • The findings link h-affine functions to precisely monotone sets and minimal hypersurfaces.
  • This work offers a foundation for further research into h-affine functions and their applications.