Nowhere differentiable intrinsic Lipschitz graphs

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. The notion of Lipschitz submanifolds in sub-Riemannian geometry was introduced, at least in the setting of Carnot groups, by B. Franchi, R. Serapioni and F. Serra Cassano in a series of seminal papers [5, 6, 7] through the theory of intrinsic Lipschitz graphs. One of the main open questions concerns the differentiability properties for such graphs: in this paper we provide examples of intrinsic Lipschitz graphs of codimension 2 (or higher) that are nowhere differentiable, i.e., that admit no homogeneous tangent subgroup at any point. Recall that a Carnot group G is a connected, simply connected and nilpotent Lie group whose Lie algebra is stratified, i.e., it can be decomposed as the direct sum ⊕j=1Vj of subspaces such that Vj+1 = [V1, Vj ] for every j = 1, . . . , s− 1, [V1, Vs] = {0}, Vs 6= {0}. We shall identify the group G with its Lie algebra via the exponential map exp : ⊕j=1Vj → G, which is a diffeomorphism. In this way, for λ > 0 one can introduce the homogeneous dilations δλ : G → G as the group automorphisms defined by δλ(p) = λ p for every p ∈ Vj . A subgroup of G is said to be homogeneous if it is dilation-invariant. Assume that a splitting G = WV of G as the product of homogeneous and complementary (i.e., such that W∩V = {0}) subgroups is fixed; we say that a function φ : W→ V intrinsic Lipschitz if there is an open nonempty cone U such that V \\ {0} ⊂ U and pU ∩ Γφ = ∅ for all p ∈ Γφ, where Γφ = {wφ(w) : w ∈W} is the intrinsic graph of φ. We say that a set Σ ⊂ G is a blow-up of Γφ at p̂ = ŵφ(ŵ) if there exists a sequence (λn)n such that λn → +∞ and the limit lim n→∞ δλn(p̂ Γφ) = Σ holds with respect to the local Hausdorff convergence. It is worth recalling that, if φ is intrinsic Lipschitz, then every blow-up is automatically the intrinsic Lipschitz graph of a map W → V. Eventually, we say that φ is intrinsically differentiable Date: January 11, 2021. 2010 Mathematics Subject Classification. 53C17, 58C20, 22E25.