A rough calculus approach to level sets in the Heisenberg group

Abstract

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in R 2 . These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a 'good calculus' on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula.

Rank-one theorem and subgraphs of BV functions in Carnot groups

“…We however point out that Theorem 1.4 is no longer valid already when k = 2 and G is the first Heisenberg group H 1 , which does not satisfy C 2 : indeed, in this setting the measure H Q−2 (Σ ⋔ ) might be either 0 or +∞ (even locally) as shown by A. Kozhevnikov [21]. See also the recent paper [25].…”

Area of intrinsic graphs and coarea formula in Carnot groups

“…is indeed a challenging open problem as soon as D H u p is surjective, see e.g. [28,30,42]. In our notation, this situation corresponds to M = {0} and L = R 2 .…”

“…s ∈ R 2 , ψ 2 ( ∩ u −1 (s)) = 0. However, a coarea formula was proved for u : H n → R 2n , assuming u to be of class C 1,α H , see [28,Theorem 6.2.5] and also [42,Theorem 8.2].…”

“…The classical coarea formula was first proved in the seminal paper [13] and it is one of the milestones of Geometric Measure Theory. Sub-Riemannian coarea formulae have been obtained in [25][26][27][33][34][35][36], assuming classical (Euclidean) regularity on the slicing function u, and in [37,42,43], assuming intrinsic regularity but only in the setting of the Heisenberg group. Here we try to work in the utmost generality: we consider a C 1 H submanifold ⊂ G, seen as the level set of a C 1 H map f with values in a stratified group M, and we slice it into level sets of a map u with values into a homogeneous group L. We will assume that L, M are complementary homogeneous subgroups of a larger homogeneous group K = LM: one could of course choose K = L × M, but our hypothesis allows to work in greater generality.…”

On exterior differential systems involving differentials of Hölder functions

“…However, recently, a growing number of applications have widened the scope of rough calculus beyond stochastics to include purely geometric problems. For instance, in [8] a rough calculus approach has been shown natural to tackle a particular case of a well-known problem of subriemannian geometry of graded nilpotent Lie groups, namely, the study of the structure of level sets of maps only intrinsically differentiable in the sense of P. Pansu [9] (such maps are known to be generically irregular in the Eucliedan sense). Namely, it has been shown that level sets of maps from the Heisenberg group H 1 to R 2 , regular only in the intrinsic sense of H 1 , are curves, possibly only Hölder regular, satisfying some "autonomous" analogue of an RDE, called in this case Level Set Differential Equation.…”