Calculus of Variations and Geometric Measure Theory

G. Antonelli - A. Merlo

Unextendable intrinsic Lipschitz curves

created by antonelli on 31 May 2021
modified on 12 Apr 2022


Accepted Paper

Inserted: 31 may 2021
Last Updated: 12 apr 2022

Journal: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
Year: 2021

ArXiv: 2105.13873 PDF


In the setting of Carnot groups, we exhibit examples of intrinisc Lipschitz curves of positive $\mathcal{H}^1$-measure that intersect every connected intrinsic Lipschitz curve in a $\mathcal{H}^1$-negligible set. As a consequence such curves cannot be extended to connected intrinsic Lipschitz curves. The examples are constructed in the Engel group and in the free Carnot group of step 3 and rank 2. While the failure of the Lipschitz extension property was already known for some pairs of Carnot groups, ours is the first example of the analogous phenomenon for intrinsic Lipschitz graphs. This is in sharp contrast with the Euclidean case.

Tags: GeoMeG