Two examples related to conical energies

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

On rectifiable measures in Carnot groups: representation

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

Plenty of big projections imply big pieces of Lipschitz graphs

I prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N . For $$1\leqslant k\leqslant n$$ 1 ⩽ k ⩽ n , we show that every intrinsic L-Lipschitz graph over a subset of a k-dimensional horizontal subgroup $${\mathbb {V}}$$ V of $${\mathbb {H}}^n$$ H n can be extended to an intrinsic $$L'$$ L ′ -Lipschitz graph over the entire subgroup $${\mathbb {V}}$$ V , where $$L'$$ L ′ depends only on L, k, and n. We further prove that 1-dimensional intrinsic 1-Lipschitz graphs in $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N , admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $${\mathbb {H}}^1$$ H 1 . The main difference to this case arises from the fact that for $$1\leqslant k<n$$ 1 ⩽ k < n , the complementary vertical subgroups of k-dimensional horizontal subgroups in $${\mathbb {H}}^n$$ H n are not commutative.