Area of intrinsic graphs and coarea formula in Carnot groups

2022/01/18 by Antoine Julia, Sebastiano Nicolussi Golo, Davide Vittone
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Geometry and complex manifolds

paper · pdf · doi:10.1007/s00209-021-02916-2

Abstract

Abstract We consider submanifolds of sub-Riemannian Carnot groups with intrinsic C¹ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math> regularity ( C¹_H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> ). Our first main result is an area formula for C¹_H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing C¹_H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> submanifolds into level sets of a C¹_H <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:math> function.

Citations

• Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

Cited by

• Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

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