Unextendable intrinsic Lipschitz curves

TL;DR

This paper constructs examples of intrinsic Lipschitz curves in Carnot groups that cannot be extended to connected curves, highlighting a fundamental difference from Euclidean geometry.

Contribution

It provides the first examples of unextendable intrinsic Lipschitz curves in Carnot groups, contrasting Euclidean extension properties.

Findings

Examples in Engel and free Carnot groups of step 3, rank 2.

Curves have positive measure and intersect all connected intrinsic Lipschitz curves negligibly.

Demonstrates failure of Lipschitz extension property in these groups.

Abstract

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.