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.

On rectifiable measures in Carnot groups: representation

“…This latter result implies that in general in item 3. of Theorem 1.1 one cannot equivalently consider as building blocks of a locally well-behaved definition of rectifiable sets the family of h-dimensional intrinsically Lipschitz graphs. So, in some sense, the result of Theorem 1.1 is sharp also in view of the negative result of [28].…”

“…We also provide an area formula for such building blocks, see Theorem 1.3. We stress that, due to the existence of intrinsic Lipschitz graphs that are nowhere intrinsically differentiable, see [28], one cannot give a geometric area formula in the spirit of Theorem 1.3 for arbitrary intrinisc Lipschitz graphs. Nevertheless the area formulae in Theorem 1.2 and Theorem 1.3 extend the area formula given in [27,Theorem 1.1], see the discussion after Theorem 1.…”

“…Nevertheless, Rademacher theorem is now known to be false in arbitrary Carnot groups in a very strict sense, i.e., there exists an h-dimensional intrinsically Lipschitz graph in a Carnot group such that at every point of it there exist infinitely many blow-ups and each of these blow-ups is not a homogeneous subgroup, see [28,Theorem 1.1]. This latter result implies that in general in item 3. of Theorem 1.1 one cannot equivalently consider as building blocks of a locally well-behaved definition of rectifiable sets the family of h-dimensional intrinsically Lipschitz graphs.…”

“…Let us stress that when a Rademacher theorem is available, one can remove the hypothesis about the intrinsic differentiability in Theorem 1.3. Nevertheless, as discussed above, a Rademacher theorem might not hold in arbitrary Carnot groups, see [28].…”

Lipschitz functions on submanifolds in Heisenberg groups

“…Fortunately, in the setting of Carnot groups intrinsic notions of rectifiability are available, modeled either on intrinsic C 1 submanifolds or on the so-called intrinsic Lipschitz graphs [15]. The two notions are in general different [17] but they coincide [25,Corollary 7.4] in Heisenberg groups H n , where intrinsic rectifiable sets are now relatively well understood and results analogue to those mentioned above are known to hold [5,6,9,10,11,13,16,19,20,21,22,25].…”