Quasiminimal crystals with a volume constraint and uniform rectifiability

AbstractWe establish here, in a quite general context, uniform rectifiability properties for quasiminimal crystals with a volume constraint. Namely we prove that to any quasiminimal crystal with a volume constraint corresponds a unique equivalent open set whose boundary is Ahlfors-regular and which satisfies the so-called condition B. Moreover implicit bounds in these properties, which imply the uniform rectifiability of the boundary, can be chosen universal. As a consequence we give a universal upper bound for the number of connected components of reduced quasiminimizers and we also prove that quasiminimal crystals with a volume constraint actually satisfy, in some universal way, an apparently stronger quasiminimality condition where admissible perturbations are not required to be volume-preserving anymore

Precisely monotone sets in step-2 rank-3 Carnot algebras

A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone. Equivalently, a precisely monotone set is a h-convex set with h-convex complement. Such sets have been introduced and classified in the Heisenberg setting by Cheeger and Kleiner in the 2010's. In the present paper, we study precisely monotone sets in the wider setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In addition to general properties, we prove a classification in step-2 rank-3 Carnot algebras that generalizes the classification already known in the Heisenberg setting using sublevel sets of h-affine functions. A significant novelty is that such sublevel sets can be different from half-spaces

Besicovitch covering property for homogeneous distances on the Heisenberg groups

We prove that the Besicovitch Covering Property (BCP) holds for homogeneous distances on the Heisenberg groups whose unit ball centered at the origin coincides with a Euclidean ball. We thus provide the first examples of homogeneous distances that satisfy BCP on these groups. Indeed, commonly used homogeneous distances, such as (Cygan-)Korányi and Carnot- Carathéodory distances, are known not to satisfy BCP. We also generalize those previous results by giving two geometric criteria that imply the non-validity of BCP and showing that in some sense our examples are sharp. To put our result in another perspective, inspired by an observation of D. Preiss, we prove that in a general metric space with an accumulation point, one can always construct bi-Lipschitz equivalent distances that do not satisfy BCP

Remarks about the besicovitch covering property in Carnot groups of step 3 and higher

We prove that the Besicovitch Covering Property (BCP) does not hold for some classes of homogeneous quasi-distances on Carnot groups of step 3 and higher. As a special case we get that, in Carnot groups of step 3 and higher, BCP is not satisfied for those homogeneous distances whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin. This result comes in contrast with the case of the Heisenberg groups where such distances satisfy BCP