Differential forms in Carnot groups: a \(\Gamma \)-convergence approach (Q662830)

A Carnot-Carathéodory (or sub-Riemannian) space is a smooth manifold \(M\) endowed with a smooth non-integrable distribution (i.e., a subbundle of the tangent bundle \(TM\)) \(H\) and a Riemannian metric on \(H\). Recall that a smooth distribution \(H\) is non-integrable if it is bracket-generating, in the sense that the iterated Lie brackets of vector fields lying in \(H\) generate the entire tangent bundle. The distribution \(H\) is usually called \textit{horizontal} and smooth paths everywhere tangent to \(H\) are referred to as \textit{horizontal paths}. The Carnot-Carathéodory (CC) distance on \(M\) is defined analogously as in Riemannian geometry as the infimum of the lengths of all horizontal paths connecting two points of \(M\). Recall that the CC-distance can be obtained as the limit of Riemannian distances with respect to metrics which penalize the motion in the direction orthogonal to \(H\) in \(TM\). In Riemannian geometry the tangent space at a point is a vector space with Euclidean structure, which serves as a good approximation to the local Riemannian geometry. Carnot-Carathéodory geometries are not locally Euclidean and for them, the closest analogue to the Riemannian tangent space is their nilpotentization. By a seminal result of \textit{J. Mitchell} [J. Differ. Geom. 21, 35--45 (1985; Zbl 0554.53023)], the Gromov-Hausdorff metric tangent cone to a Carnot-Carathéodory manifold at a regular point \(q\) exists and is the Carnot group arising from the nilpotentization of \(H\) at \(q\). This result was later extended to non-regular points by Bellaiche; see [\textit{R. Montgomery}, A tour of subriemannian geometries, their geodesics and applications. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1044.53022)]. Carnot groups therefore serve as rigid (since they are invariant under left translations and group dilations) tangent spaces to Carnot-Carathéodory spaces. A Carnot group \(\mathbb{G}\) can be thought of as the Lie group \((\mathbb{R}^n, \cdot)\), where \(\cdot\) is a (non-commutative) multiplication, such that the Lie algebra \(\mathfrak{g}\) of \(\mathbb{G}\) admits a step \(k\) stratification: \[ \mathfrak{g} = V_1 \oplus \cdots \oplus V_k, \] where \([V_1,V_i] = V_{i+1}\), \(V_k \neq \{ 0 \}\) and \(V_i = \{ 0 \}\), for \(i > k\). The simplest example of a Carnot group is the first Heisenberg group \(\mathbb{H}^1\), which as a set can be identified with \(\mathbb{R}^3\), with coordinates \((x,y,t)\) and the stratification of the Lie algebra is given by \(\mathfrak{g} = V_1 \oplus V_2\), where \(V_1 = \text{span} \{ \partial_x - \frac{1}{2} y \partial_t, \partial_y + \frac{1}{2} x \partial_x \}\) and \(V_2 = \text{span}\{ \partial_t \}\). The paper under review is concerned with so called intrinsic differential forms on Carnot groups. A 1-form on a Carnot group is intrinsic if it is dual to an intrinsic vector field (which is just a section of the horizontal bundle). The problem of defining an intrinsic exterior differential on a Carnot group is a subtle one. It was recently solved by Rumin, who defined a subcomplex \((E^\ast_0,d_c)\) of the de Rham complex such that its 1-forms are exactly the intrinsic ones (in the above sense), the intrinsic differential \(d_c f\) of a smooth function is its horizontal differential, and furthermore, the complex \((E_0^\ast,d_c)\) is exact and self-dual under the Hodge \(\ast\)-duality. The main result of the present paper is that the intrinsic differential \(d_c\) is a natural limit of the usual (properly weighted) Riemannian differentials \(d_\varepsilon\). More precisely, the authors show that on a free Carnot group of step \(k\), the \(L^2\)-energy associated with \(\varepsilon^{-k} d_\varepsilon\) on 1-forms converge to the associated \(L^2\)-energy of \(d_c\), where the convergence is De Georgi's \(\Gamma\)-convergence.