Contact sub-Riemannian Manifolds Research Articles

Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces

The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $\mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of $\mathbb {R}^n$ —where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.

Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold

We construct and study the intrinsic sub-Laplacian, defined outside the set of characteristic points, for a smooth hypersurface embedded in a contact sub-Riemannian manifold. We prove that, away from characteristic points, the intrinsic sub-Laplacian arises as the limit of Laplace–Beltrami operators built by means of Riemannian approximations to the sub-Riemannian structure using the Reeb vector field. We carefully analyse three families of model cases for this setting obtained by considering canonical hypersurfaces embedded in model spaces for contact sub-Riemannian manifolds. In these model cases, we show that the intrinsic sub-Laplacian is stochastically complete and in particular, that the stochastic process induced by the intrinsic sub-Laplacian almost surely does not hit characteristic points.

Steiner and tube formulae in 3D contact sub-Riemannian geometry

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in [Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi and K. Wildrick, Steiner’s formula in the Heisenberg group, Nonlinear Anal. 126 (2015) 201–217; M. Ritoré, Tubular neighborhoods in the sub-Riemannian Heisenberg groups, Adv. Calc. Var. 14(1) (2021) 1–36] for the Heisenberg group.

Lipschitz regularity for solutions of the parabolic p-Laplacian in the Heisenberg group

We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic \(p\)-Laplacian \(\partial_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),\)in a cylinder \(\Omega\times\mathbb{R}^+\), where \(\Omega\) is domain in the Heisenberg group \(\mathbb{H}^n\), and \(2\le p \le 4\). The result continues to hold in the more general setting of contact subRiemannian manifolds.

Surfaces of genus g ≥ 1 in 3D contact sub-Riemannian manifolds

We consider smooth embedded surfaces in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the fmiteness/not-finiteness of the distance.

On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.

A de Rham decomposition type theorem for contact sub-Riemannian manifolds

In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field xi is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point qin M such that the holonomy group Psi (q) acts reducibly on H(q) yielding a decomposition H(q) = H_1(q)oplus cdots oplus H_m(q) into Psi (q)-irreducible factors. Using parallel transport we obtain the decomposition H = H_1oplus cdots oplus H_m of H into sub-distributions H_i. Unlike the Riemannian case, the distributions H_i are not integrable, however they induce integrable distributions Delta _i on M/xi , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/xi )=Delta _1oplus cdots oplus Delta _m, and the latter decomposition of T(U/xi ) induces the decomposition of U/xi into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.

Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds

On considère des processus stochastiques sur des surfaces dans des espaces tridimensionnels de contact sous-riemanniens. En utilisant des approximations riemanniennes définies par le champ de Reeb, on obtient un opérateur différentiel du second ordre sur la surface comme limite des opérateurs de Laplace–Beltrami correspondants. Le processus stochastique associé à l’opérateur limite est défini le long de la foliation caractéristique induite sur la surface par la distribution de contact. Nous montrons que pour ce processus stochastique, les points elliptiques sont inaccessibles alors que les points hyperboliques sont accessibles à travers les séparatrices. On discute les résultats sur des exemples et on identifie des surfaces canoniques dans le groupe de Heisenberg, ainsi que dans les groupes SU(2) et SL(2,R) équipés de leurs structures sous-riemanniennes de contact canoniques, jouant le rôle de cas modèles dans ce contexte. Ces techniques nous permettent de plus de dériver une expression de la courbure gaussienne intrinsèque d’une surface générale dans une variété sous-riemannienne de dimension trois.

Short Geodesics Losing Optimality in Contact Sub-Riemannian Manifolds and Stability of the 5-Dimensional Caustic

We study the sub-Riemannian exponential for contact distributions on manifolds of dimension greater or equal to 5. We compute an approximation of the sub-Riemannian Hamiltonian flow and show that the conjugate time can have multiplicity 2 in this case. We obtain an approximation of the first conjugate locus for small radii and introduce a geometric invariant to show that the metric for contact distributions typically exhibits an original behavior, different from the classical 3-dimensional case. We apply these methods to the case of 5-dimensional contact manifolds. We provide a stability analysis of the sub-Riemannian caustic from the Lagrangian point of view and classify the singular points of the exponential map.

Conformality and Q-harmonicity in sub-Riemannian manifolds

We establish regularity of conformal maps between sub-Riemannian manifolds from regularity of Q-harmonic functions, and in particular we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth in all contact sub-Riemannian manifolds. Together with the recent results in [15], our work yields a new proof of the smoothness of boundary extensions of biholomorphims between strictly pseudoconvex smooth domains [29].

Curvatures of homogeneous sub-Riemannian manifolds

The author proved in the late 1980s that any homogeneous manifold with an intrinsic metric is isometric to some homogeneous quotient space of a connected Lie group by its compact subgroup with an invariant Finslerian or sub-Finslerian metric. In the case of a trivial compact subgroup, the invariant Riemannian or sub-Riemannian metrics are singled out from invariant Finslerian or sub-Finslerian metrics by being in one-to-one correspondence with special one-parameter Gaussian convolutions semigroups of absolutely continuous probability measures. Any such semigroup is generated by a second order hypoelliptic operator. In the present paper, in connection with this, the author discusses briefly the operator definition of lower bound for Ricci curvature by Baudoin–Garofalo. Earlier, Agrachev defined a notion of curvature for sub-Riemannian manifolds. As an alternative, the author discusses in some detail old definitions of curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov’ev. To calculate the Solov’ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant bracket generating distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin–Hamilton function. This function was applied in the Pontryagin maximum principle for the time-optimal problem. The foliation is entirely described by the coadjoint representation of the Lie group G. We also use the canonical symplectic form on and its values for the above mentioned invariant Hamiltonian vector fields. In particular, the above rigging method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds possessing a submetry onto a Riemannian manifold. In Sects. 5 and 6, some examples are presented.

On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures

In this paper 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 A3 and A5 (in the classification of Arnol’d’s school). We show that in the non-generic case, a cornucopia of asymptotics can occur, even for Riemannian surfaces. MSC classes: 53C17 · 57R45 · 58J35

The regularity of Euclidean Lipschitz boundaries with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds

In this paper we consider a set E⊂Ω with prescribed mean curvature f∈C(Ω) and Euclidean Lipschitz boundary ∂E=Σ inside a three-dimensional contact sub-Riemannian manifold M. We prove that if Σ is locally a regular intrinsic graph, the characteristic curves are of class C2. The result is shape and improves the ones contained in Capogna et al. (2009) and Galli and Ritoré (2015).

Sub-Riemannian Curvature in Contact Geometry

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.

Regularity of $$C^1$$ C 1 surfaces with prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds

In this paper we consider surfaces of class \(C^1\) with continuous prescribed mean curvature in three-dimensional contact sub-Riemannian manifolds and prove that their characteristic curves are of class \(C^2\). This regularity also holds for critical points of the sub-Riemannian perimeter under a volume constraint. All results are valid in the first Heisenberg group \({\mathbb {H}}^1\).

How many geodesics join two points on a contact sub-Riemannian manifold?

We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordinates $(x,z) \in \mathbb{R}^{2n} \times \mathbb{R}$ centered at $p$. In this case we prove that for the generic $q$ the number of geodesics $\nu(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{|z|}{\|x\|^2} + R_1 \leq \nu(q) \leq C_2\frac{|z|}{\|x\|^2} + R_2\] for some constants $C_1,C_2$ and $R_1,R_2$. We recover exact values for Heisenberg groups, where $C_1=C_2 = \frac{8}{\pi}$. Removing the genericity condition for $q$, geodesics might appear in families and we prove a similar statement for their topology. We study these families, and in particular we focus on the unexpected appearance of isometrically non-equivalent geodesics: families on which the action of isometries is not transitive. We apply the previous study to contact sub-Riemannian manifolds: we prove that for any given point $p \in M$ there is a sequence of points $p_n$ such that $p_n \to p$ and that the number of geodesics between $p$ and $p_n$ grows unbounded (moreover these geodesics have the property of being contained in a small neighborhood of $p$).

Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

Bishop and Laplacian Comparison Theorems on Three-Dimensional Contact Sub-Riemannian Manifolds with Symmetry

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three-dimensional contact sub-Riemannian manifolds with symmetry.

Existence of isoperimetric regions in contact sub-Riemannian manifolds

We prove existence of regions minimizing perimeter under a volume constraint in contact sub-Riemannian manifolds whose quotient by the group of contact transformations preserving the sub-Riemannian metric is compact.

Sur les variétés sous-riemanniennes de contact isotropes

In this Note, we show that contrary to the dimension 3 case, isotropic contact sub-Riemannian manifolds of dimension greater than 3 do not exist. To cite this article: A.-R. Mansouri, C. R. Acad. Sci. Paris, Ser. I 339 (2004).