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

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## Abstract

We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace–Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in SU(2) and SL(2,R) equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.

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... The work [BBC22] analyses the metric structure, particularly near characteristic points, induced on surfaces embedded in three-dimensional contact sub-Riemannian manifolds, and [BBCH21] introduces and studies properties of a canonical stochastic process on surfaces in three-dimensional contact sub-Riemannian manifolds which exhibits different behaviours near an elliptic characteristic point and a hyperbolic characteristic point. ...
... The present article aims to initiate further studies of hypersurfaces embedded in higher-dimensional contact sub-Riemannian manifolds. We intrinsically construct a sub-Laplacian on hypersurfaces in contact sub-Riemannian manifolds, which for surfaces in three-dimensional contact sub-Riemannian manifolds gives rise to the generator of the stochastic process obtained in [BBCH21] by means of Riemannian approximations, and we use our analysis to propose model cases for this setting. Some notions such as horizontal connectivity, horizontal connection and horizontal mean curvature on hypersurfaces in sub-Riemannian manifolds are studied by Tan and Yang [TY04]. ...
... We then state the result that it emerges as the limit of Laplace-Beltrami operators. This particularly implies that the operator ∆ 0 constructed in [BBCH21] on surfaces in three-dimensional contact sub-Riemannian manifolds coincides with the intrinsic sub-Laplacian defined in this article for the case n = 1. ...
Preprint
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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.
... If in the construction of the Laplace-Beltrami operator instead of the Riemannian volume we take any smooth volume, then there is no singularity present (except the degeneracy of the principal symbol) and such operators can be handled using essentially the theory of Hörmander operators even though the singularity still manifests itself in various forms [20,60]. We should also mention separately articles [9,55], where some results concerning analysis of some structures with similar singularities were obtained. In [55] the authors studied the heat content on domains with characteristic points, while in [9] induced stochastic processes on surfaces in the Heisenberg group are studied. ...
... We should also mention separately articles [9,55], where some results concerning analysis of some structures with similar singularities were obtained. In [55] the authors studied the heat content on domains with characteristic points, while in [9] induced stochastic processes on surfaces in the Heisenberg group are studied. ...
Article
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The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.
... Gauge balls are actually characterized by such a weighted mean value property for ∆ H n -harmonic functions as proved by Lanconelli in [23]. The metric balls B R defined in (3) are not the unique choice of "balls" adapting to the subRiemannian features of the Heisenberg group. For instance, the Carnot-Carathédory balls play somehow the role of the geodesic balls in H n . ...
... In this section we collect some preliminary material that will be used in the rest of the paper. We shall recall some known notions for the study of smooth hypersurfaces in H n , and we refer the reader to [11,34,32,7,12,35,20,37,10,2,9,3] for several insights and different perspectives and approaches to the geometry of submanifolds in various subRiemannian settings. Being ·, · the metric defined in the Introduction (with induced norm | · |), we denote by ∇ the Levi-Civita connection associated to this metric. ...
Article
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In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces.
... Gauge balls are actually characterized by such a weighted mean value property for ∆ H n -harmonic functions as proved by Lanconelli in [23]. The metric balls B R defined in (3) are not the unique choice of "balls" adapting to the subRiemannian features of the Heisenberg group. For instance, the Carnot-Carathédory balls play somehow the role of the geodesic balls in H n . ...
... In this section we collect some preliminary material that will be used in the rest of the paper. We shall recall some known notions for the study of smooth hypersurfaces in H n , and we refer the reader to [11,34,32,7,12,35,20,37,10,2,9,3] for several insights and different perspectives and approaches to the geometry of submanifolds in various subRiemannian settings. Being ·, · the metric defined in the Introduction (with induced norm | · |), we denote by ∇ the Levi-Civita connection associated to this metric. ...
Preprint
Full-text available
In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces
... We introduce the horizontal Hessian for classifying characteristic points (cf. also [BBCH20]). Fix an affine connection ∇ on the distribution D. Then, the horizontal Hessian of u ∈ C 2 (M ) is the (0, 2)-tensor on D, defined by ...
... As was the case for the horizontal mean curvature, the definition of K 0 restricted at Σ is independent on the choice of the local defining function. The intrinsic Gaussian curvature has been introduced in [BTV17] for the Heisenberg group and in [BBCH20] for the general 3D contact case. We refer to those articles for further details. ...
Thesis
Sub-Riemannian geometry is a particularly rich class of metric structures, which generalizes Riemannian geometry, where a smoothly varying metric is defined only on a subset of preferred directions of the tangent space at each point of a smooth manifold M (called horizontal directions). Under the so-called Hörmander condition, M is horizontally-path connected, and the usual length-minimization procedure yields a well-defined metric. TheLaplace-Beltrami operator is generalized by the sub-Laplacian which is subelliptic, but has nonetheless suitable regularity properties (in particular, it is hypoelliptic). In this thesis, we investigate the heat content asymptotics and related topics in sub-Riemannian geometry.
... In addition, we show that for "model" surfaces in the model spaces SU (2) and SL(2) (cf. [BBCH21], see also [BH22]), it holds a 2 = a 3 = 0 as for the horizontal plane in H. ...
Preprint
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 10.1016/j.na.2015.05.006 and arXiv:1703.01592v3 for the Heisenberg group.
... If we restrict ourselves to a subsurface Σ ⊆ M , then the picture is quite different. If h ε is the induced metric on Σ from g ε , then d hε does not converge to a metric compatible with the topology if it converges at all, see [3,4] for details. Seeing that this limit breaks the topology of Σ, it is interesting to study the limit of the Gauss-Bonnet formula with respect to h ε as ε ↓ 0. See [5,7] for previous results relating topology to sub-Riemannian invariants. ...
Preprint
We obtain a sub-Riemannian version of the classical Gauss-Bonnet theorem. We consider 3 dimensional contact sub-Riemannian manifolds with a corresponding metric extension and obtain a pure sub-Riemannian metric in the limit for subsurfaces. In particular, we are able to recover topological information concentrated around the characteristic set of points where the tangent space to the surface and contact structure coincide.
... We introduce the horizontal Hessian for classifying characteristic points (cf. also [BBCH20]). Fix an affine connection∇ on the distribution D. Then, the horizontal Hessian of u ∈ C 2 (M ) is the (0, 2)-tensor on D, defined as ...
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J. M. M. Veloso. Limit of Gaussian and normal curvatures of surfaces in Riemannian approximation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet theorem, 2020. Available at arXiv:2002.07177.