On the Hausdorff volume in sub-Riemannian geometry

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4 on every smooth curve) but in general not C^5. These results answer to a question addressed by Montgomery about the relation between two intrinsic volumes that can be defined in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent approximation depends on the point (that may happen starting from dimension 5), then they are not proportional, in general.Comment: Accepted on Calculus and Variations and PD

On 2-step, corank 2 nilpotent sub-Riemannian metrics

In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric

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

On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $\Delta_\omega$, as the divergence of the horizontal gradient, once a volume $\omega$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of random walks, where \emph{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 $L^c$. 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 $P$) 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 $c$ such that $\Delta_\omega=L^c$. On Carnot groups, if $H$ is the Haar volume, then there always exists a complement $c$ such that $\Delta_H=L^c$. However this complement is not unique in general. For quasi-contact structures, in general, $\Delta_P \neq L^c$ for any choice of $c$. In particular, $L^c$ is not symmetric w.r.t. Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, $\Delta_P$ 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.Comment: 42 pages, 1 figure. v2: minor revisions; v3: minor typos corrected; v4: final version, to appear on Advances in Mathematic

Invariants, volumes and heat kernels in sub-Riemannian geometry

Sub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34]) Very recently, it appeared in the eld of cognitive neuroscience to model the functional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35])

Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry

Hausdorff volume in non equiregular sub-Riemannian manifolds

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give conditions under which it is a Radon measure. We finally give a complete characterization of the singular part. We illustrate our results and techniques on numerous examples and cases (e.g. to generic sub-Riemannian structures)

Chemoenzymatic synthesis and pharmacological characterization of functionalized aspartate analogues as novel excitatory amino acid transporter inhibitors

Aspartate (Asp) derivatives are privileged compounds for investigating the roles governed by excitatory amino acid transporters (EAATs) in glutamatergic neurotransmission. Here, we report the synthesis of various Asp derivatives with (cyclo)alkyloxy and (hetero)aryloxy substituents at C-3. Their pharmacological properties were characterized at the EAAT1-4 subtypes. The L-threo-3-substituted Asp derivatives 13a-e and 13g-k were non-substrate inhibitors, exhibiting pan activity at EAAT1-4 with IC50 values ranging from 0.49 to 15 μM. Comparisons between (DL-threo)-19a-c and (DL-erythro)-19a-c Asp analogues confirmed that the threo configuration is crucial for the EAAT1-4 inhibitory activities. Analogues (3b-e) of L-TFB-TBOA (3a) were shown to be potent EAAT1-4 inhibitors, with IC50 values ranging from 5-530 nM. Hybridization of the nonselective EAAT inhibitor L-TBOA with EAAT2-selective inhibitor WAY-213613 or EAAT3-preferring inhibitor NBI-59159 yielded compounds 8 and 9, respectively, which were non-selective EAAT inhibitors displaying considerably lower IC50 values at EAAT1-4 (11-140 nM) than those displayed by the respective parent molecules