Calculus of Variations and Geometric Measure Theory
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U. Boscain - R. Neel - L. Rizzi

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

created by rizzi1 on 08 Feb 2016


Submitted Paper

Inserted: 8 feb 2016
Last Updated: 8 feb 2016

Year: 2015
Links: arXiv preprint


On a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian $\Delta_\omega$, as the divergence of the horizontal gradient, once a volume $\omega$ is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. We consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement $\mathbf{c}$ to the sub-Riemannian distribution, and is denoted $L^{\mathbf{c}}$.

We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one $\mathcal{P}$) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation. On contact structures, for every volume $\omega$, there exists a unique complement $\mathbf{c}$ such that $\Delta_\omega= L^{\mathbf{c}}$. On Carnot groups, if $H$ is the Haar volume, then there always exists a complement $\mathbf{c}$ such that $\Delta_H = L^{\mathbf{c}}$. However this complement is not unique in general. For quasi-contact structures, in general, $\Delta_{\mathcal{P}} \neq L^{\mathbf{c}}$ for any choice of $\mathbf{c}$. In particular, $L^{\mathbf{c}}$ is not symmetric w.r.t. Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, $\Delta_{\mathcal{P}}$ is the unique intrinsic macroscopic Laplacian.

A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above.


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