sub-Riemannian Geometry Research Articles

Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by f(x)=x3 near its critical point x=0. As a consequence, we characterise conjugate points in ideal sub-Riemannian manifolds in terms of the metric structure of the space. The proof uses the Hilbert invariant integral of the associated variational problem.

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.

Surface measure on, and the local geometry of, sub-Riemannian manifolds

We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds. Among various technical tools, we establish a general criterion for the uniform convergence of parametrized sub-Riemannian distances, and local uniform asymptotics for the diameter of small metric balls.

Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

The Sub-Riemannian Geometry of Screw Motions with Constant Pitch

The Sub-Riemannian Geometry of Screw Motions with Constant Pitch

Nodal Sets of Eigenfunctions of Sub-Laplacians

Abstract Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians. A standard example is the sum of squares of bracket-generating vector fields on compact quotients of the Heisenberg group. Our results show that nodal sets behave in an anisotropic way, which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation, and desingularization at singular points. Furthermore, we provide a simple example demonstrating that for sub-Laplacians, the Hausdorff measure of nodal sets of eigenfunctions cannot be bounded above by $\sqrt \lambda $, which is the bound conjectured by Yau for Laplace–Beltrami operators on smooth manifolds.

Subdifferentials and minimizing Sard conjecture in sub-Riemannian geometry

Subdifferentials and minimizing Sard conjecture in sub-Riemannian geometry

Multiband Image Fusion via Regularization on a Riemannian Submanifold

Multiband image fusion aims to generate high spatial resolution hyperspectral images by combining hyperspectral, multispectral or panchromatic images. However, fusing multiband images remains a challenge due to the identifiability and tracking of the underlying subspace across varying modalities and resolutions. In this paper, an efficient multiband image fusion model is proposed to investigate the latent structures and intrinsic physical properties of a multiband image, which is characterized by the Riemannian submanifold regularization method, nonnegativity and sum-to-one constraints. An alternating minimization scheme is proposed to recover the latent structures of the subspace via the manifold alternating direction method of multipliers (MADMM). The subproblem with Riemannian submanifold regularization is tackled by the projected Riemannian trust-region method with guaranteed convergence. The effectiveness of the proposed method is demonstrated on two multiband image fusion problems: (1) hyperspectral and panchromatic image fusion and (2) hyperspectral, multispectral and panchromatic image fusion. The experimental results confirm that our method demonstrates superior fusion performance with respect to competitive state-of-the-art fusion methods.

The Geometry of Riemannian Curvature Radii

We study the geometric structures associated with curvature radii of curves with values on a Riemannian manifold (M, g). We show the existence of sub-Riemannian manifolds naturally associated with the curvature radii and we investigate their properties. In the particular case of surfaces these sub-Riemannian structures are of Engel type. The main character of our construction is a pair of global vector fields f1,f2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_1,f_2$$\\end{document}, which encodes intrinsic information on the geometry of (M, g).

On the dimension of the algebras of local infinitesimal isometries of 3-dimensional special sub-Riemannian manifolds

Suppose that we are given a contact sub-Riemannian manifold (M, H, g) of dimension 3 such that the Reeb vector field is an infinitesimal isometry (such manifolds will be referred to as special). For a point qin M denote by mathfrak {i}^*(q) the Lie algebra of germs at q of infinitesimal isometries of (M, H, g). It is proved that for a generic point qin M, dim mathfrak {i}^*(q) can only assume the values 1, 2, 4. Moreover, dim mathfrak {i}^*(q) = 4 if and only if the curvature function determined by the canonical sub-Riemannian connection is constant in a neighborhood of q. The latter case is possible if (M, H, g) is locally isometric, in a neighborhood of q, to a left-invariant sub-Riemannian structure on a 3-dimensional Lie group.

Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian,12Δ(‖∇u‖2)≥g(∇u,∇Δu)+K‖∇u‖2,K∈R, implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry–Émery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting.

Heat Kernels, Stochastic Processes and Functional Inequalities

The workshop provided a forum for recent progress on a wide array of topics at the nexus of Analysis (elliptic, subelliptic and parabolic differential equations), Geometry (Riemannian and sub-Riemannian geometries, metric measure spaces, geometric analysis and curvature), and Probability Theory (Brownian motion, Dirichlet spaces, stochastic calculus and random media). The workshop provides a unique opportunity to encourage and foster interactions between mathematicians who share some common interests but might use different research tools or work in different mathematical settings.

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.

Parallel Codazzi tensors with submanifold applications

AbstractA decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

This paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper, we proved an abstract result and discussed operators on the Heisenberg group. Here, we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and 0-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

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.

Removable sets and [formula omitted]-uniqueness on manifolds and metric measure spaces

We study symmetric diffusion operators on metric measure spaces. Our main question is whether essential self-adjointness or Lp-uniqueness are preserved under the removal of a small closed set from the space. We provide characterizations of the critical size of removed sets in terms of capacities and Hausdorff dimension without any further assumption on removed sets. As a key tool we prove a non-linear truncation result for potentials of nonnegative functions. Our results are robust enough to be applied to Laplace operators on general Riemannian manifolds as well as sub-Riemannian manifolds and metric measure spaces satisfying curvature-dimension conditions. For non-collapsing Ricci limit spaces with two-sided Ricci curvature bounds we observe that the self-adjoint Laplacian is already fully determined by the classical Laplacian on the regular part.

Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold.

We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner's formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner's formula follows a generalized second-order calculus of Kullback-Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric.

Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of “good” cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.

Homogenization for sub-riemannian Lagrangians

We consider Lagrangians that are defined only on the horizontal distribution of a sub-riemannian manifold. The associated Hamiltonian is neither strictly convex nor coercive. We prove a result on homogenization of the Hamilton–Jacobi equation following the approach in Contreras et al (2015 Calc. Var. Partial Differ. Equ. 52 237–52). To establish the result, we need to extend weak KAM and Aubry–Mather theories to the sub-riemannian setting. We obtain a Tonelli theorem to dispend of the Lagrangian dynamics tools used in standard weak KAM theory. In the way towards our main result we observe that, the long time convergence of the Lax-Oleinik semigroup, and the vanishing discount convergence of the discounted value function hold.