Calculus of Variations and Geometric Measure Theory

G. Antonelli - D. Di Donato - S. Don - E. Le Donne

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

created by didonato on 26 May 2020
modified by antonelli on 11 Jan 2022


Accepted Paper

Inserted: 26 may 2020
Last Updated: 11 jan 2022

Journal: Annales de l'Institut Fourier
Year: 2020

ArXiv: 2005.11390 PDF


In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of H\"older properties along the projections of left-invariant vector fields on the graph. We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group. As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.

Tags: GeoMeG