Section: New Results
New results: geometric control
Let us list some new results in subRiemannian geometry and hypoelliptic diffusion obtained by GECO's members.

In [2] we compare different notions of curvature on contact subRiemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the subRiemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the subRiemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a subRiemannian version of the BonnetMyers theorem that applies to any contact manifold.

In [3] we provide the smalltime heat kernel asymptotics at the cut locus in three relevant cases: generic lowdimensional Riemannian manifolds, generic 3D contact subRiemannian manifolds (close to the starting point) and generic 4D quasicontact subRiemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic lowdimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are ${A}_{3}$ and ${A}_{5}$ (in Arnol'd's classification). We show that in the nongeneric case, a cornucopia of asymptotics can occur, even for Riemannian surfaces.

In [5] we study the evolution of the heat and of a free quantum particle (described by the Schrödinger equation) on twodimensional manifolds endowed with the degenerate Riemannian metric $d{s}^{2}=d{x}^{2}+{\leftx\right}^{2\alpha}d{\theta}^{2}$, where $x\in \mathbb{R}$, $\theta \in {S}^{1}$ and the parameter $\alpha \in \mathbb{R}$. For $\alpha \le 1$ this metric describes conelike manifolds (for $\alpha =1$ it is a flat cone). For $\alpha =0$ it is a cylinder. For $\alpha \ge 1$ it is a Grushinlike metric. We show that the LaplaceBeltrami operator $\Delta $ is essentially selfadjoint if and only if $\alpha \notin (3,1)$. In this case the only selfadjoint extension is the Friedrichs extension ${\Delta}_{F}$, that does not allow communication through the singular set $\{x=0\}$ both for the heat and for a quantum particle. For $\alpha \in (3,1]$ we show that for the Schrödinger equation only the average on $\theta $ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is ${\Delta}_{F}$) cannot. For $\alpha \in (1,1)$ we prove that there exists a canonical selfadjoint extension ${\Delta}_{N}$, called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the ${L}^{1}$ norm for the heat equation) of the Markovian extensions ${\Delta}_{F}$ and ${\Delta}_{B}$, proving that ${\Delta}_{F}$ is stochastically complete at the singularity if and only if $\alpha \le 1$, while ${\Delta}_{B}$ is always stochastically complete at the singularity.

In [6] we study spectral properties of the Laplace–Beltrami operator on two relevant almostRiemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. As for general almostRiemannian structures (under certain technical hypothesis), the singular set acts as a barrier for the evolution of the heat and of a quantum particle, although geodesics can cross it. This is a consequence of the selfadjointness of the Laplace–Beltrami operator on each connected component of the manifolds without the singular set. We get explicit descriptions of the spectrum, of the eigenfunctions and their properties. In particular in both cases we get a Weyl law with dominant term $ElogE$. We then study the effect of an AharonovBohm nonapophantic magnetic potential that has a drastic effect on the spectral properties. Other generalized Riemannian structures including conic and anticonic type manifolds are also studied. In this case, the AharonovBohm magnetic potential may affect the selfadjointness of the LaplaceBeltrami operator.

Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In [7] we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with respect to a given conformal structure. The singularities correspond to the zeros of the vector fields and the genericity is considered with respect to a natural topology in the space of pairs of vector fields. Line fields at generic singularities turn out to be topologically equivalent to the Lemon, Star and Monstar singularities that one finds at umbilical points.

In [10] we prove that any corank 1 Carnot group of dimension $k+1$ equipped with a leftinvariant measure satisfies the measure contraction property $\mathrm{MCP}(K,N)$ if and only if $K\le 0$ and $N\ge k+3$. This generalizes the well known result by Juillet for the Heisenberg group ${H}^{k+1}$ to a larger class of structures, which admit nontrivial abnormal minimizing curves. The number $k+3$ coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least $N$ such that the $\mathrm{MCP}(0,N)$ is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat.

In [14] we relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and subRiemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on subRiemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of subLaplacian on a general subRiemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.

By adapting a technique of Molchanov, we obtain in [15] the heat kernel asymptotics at the subRiemannian cut locus, when the cut points are reached by a $r$dimensional parametric family of optimal geodesics. We apply these results to the biHeisenberg group, that is, a nilpotent leftinvariant subRiemannian structure on ${\mathbb{R}}^{5}$ depending on two real parameters ${\alpha}_{1}$ and ${\alpha}_{2}$. We develop some results about its geodesics and heat kernel associated to its subLaplacian and we point out some interesting geometric and analytic features appearing when one compares the isotropic (${\alpha}_{1}={\alpha}_{2}$) and the nonisotropic cases (${\alpha}_{1}\ne {\alpha}_{2}$). In particular, we give the exact structure of the cut locus, and we get the complete smalltime asymptotics for its heat kernel.

The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to several settings, among which the one of Carnot groups. However, the target space has generally been assumed to be equal to ${\mathbb{R}}^{d}$ for some $d\ge 1$. We focus in [17] on the extendability problem for general ordered pairs $({G}_{1},{G}_{2})$ (with ${G}_{2}$ nonAbelian). We analyze in particular the case ${G}_{1}=\mathbb{R}$ and characterize the groups ${G}_{2}$ for which the Whitney extension property holds, in terms of a newly introduced notion that we call pliability. Pliability happens to be related to rigidity as defined by Bryant an Hsu. We exploit this relation in order to provide examples of nonpliable Carnot groups, that is, Carnot groups so that the Whitney extension property does not hold. We use geometric control theory results on the accessibility of control affine systems in order to test the pliability of a Carnot group.

In [19] we study the cut locus of the free, step two Carnot groups ${G}^{k}$ with $k$ generators, equipped with their leftinvariant Carnot–Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in the literature, by exhibiting sets of cut points $C\subset {G}^{k}$ which, for $k\ge 4$, are strictly larger than conjectured ones. Furthermore, we study the relation of the cut locus with the socalled abnormal set. For each $k\ge 4$, we show that, contrarily to the case $k=2,3$, the cut locus always intersects the abnormal set, and there are plenty of abnormal geodesics with finite cut time. Finally, and as a straightforward consequence of our results, we derive an explicit lower bound for the small time heat kernel asymptotics at the points of $C$. The question whether $C$ coincides with the cut locus for $k\ge 4$ remains open.
We also edited the two volumes [13] and [12], containing some of the lecture notes of the courses given during the IHP triemster on “Geometry, Analysis and Dynamics on subRiemannian Manifolds” which we organized in Fall 2014. The second volume also contains a chapter [11] coauthored by members of the team.