Nowhere differentiable intrinsic Lipschitz graphs

TL;DR

This paper constructs intrinsic Lipschitz graphs in Carnot groups that have infinitely many distinct blow-up limits at every point, challenging the applicability of Rademacher's theorem in this setting.

Contribution

It provides the first examples of intrinsic Lipschitz graphs with non-unique blow-up limits, showing limitations of differentiability theorems in Carnot groups.

Findings

Existence of intrinsic Lipschitz graphs with infinitely many blow-up limits

Counterexamples to Rademacher's theorem in Carnot groups

Demonstration of non-differentiability phenomena in sub-Riemannian geometry

Abstract

We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.