Section: New Results

New results: geometric control

Let us list some new results in sub-Riemannian geometry and hypoelliptic diffusion obtained by GECO's members.

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

• In [3] we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional 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 non-generic 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 two-dimensional manifolds endowed with the degenerate Riemannian metric $d{s}^2=d{x}^2+{\left|x\right|}^{-2\alpha }d{\theta }^2$, where $x\in ℝ$, $\theta \in S^1$ and the parameter $\alpha \in ℝ$. For $\alpha \le -1$ this metric describes cone-like manifolds (for $\alpha =-1$ it is a flat cone). For $\alpha =0$ it is a cylinder. For $\alpha \ge 1$ it is a Grushin-like metric. We show that the Laplace-Beltrami operator $\Delta$ is essentially self-adjoint if and only if $\alpha \notin (-3,1)$. In this case the only self-adjoint 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 self-adjoint 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 almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. As for general almost-Riemannian 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 self-adjointness 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 Aharonov-Bohm non-apophantic magnetic potential that has a drastic effect on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami 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 left-invariant 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 non-trivial 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 sub-Riemannian 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 sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian 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 sub-Riemannian cut locus, when the cut points are reached by a $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Riemannian structure on ${ℝ}^5$ depending on two real parameters ${\alpha }_1$ and ${\alpha }_2$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we point out some interesting geometric and analytic features appearing when one compares the isotropic (${\alpha }_1={\alpha }_2$) and the non-isotropic cases (${\alpha }_1\ne {\alpha }_2$). In particular, we give the exact structure of the cut locus, and we get the complete small-time 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 ${ℝ}^d$ for some $d\ge 1$. We focus in [17] on the extendability problem for general ordered pairs $(G_1,G_2)$ (with $G_2$ non-Abelian). We analyze in particular the case $G_1=ℝ$ 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 non-pliable 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 left-invariant 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 so-called 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 sub-Riemannian Manifolds” which we organized in Fall 2014. The second volume also contains a chapter [11] co-authored by members of the team.