Note shows Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
problem Proving Cheng-Yau gradient estimate for specific geometric structures.
method Utilizing previous results on sub-Riemannian manifolds and Carnot groups.
result Established Cheng-Yau estimate for Carnot groups and sub-Riemannian manifolds.
Maps commuting with sub-Laplacians on Carnot groups are conformal.
problem Characterizing maps preserving sub-Laplacians on sub-Riemannian Lie groups.
method Analyzing smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians.
result Sub-Laplacian determines the sub-Riemannian structure in Carnot groups.
Proves measure contraction for specific sub-Riemannian structures.
problem Measure contraction properties in sub-Riemannian structures.
method Analytic sub-Riemannian structures and Lipschitz Carnot groups.
result Proves measure contraction properties for the structures.
The paper studies geodesics in Carnot groups and their properties.
problem Understanding geodesics in sub-Riemannian and sub-Finsler manifolds.
method Analyzing infinitesimal and asymptotic properties of geodesics in Carnot groups.
result Blowups of geodesics in Carnot groups are still geodesics in lower rank groups.
Proves a theorem in sub-Riemannian geometry using Carnot groups.
problem Proving a maximum modulus theorem in sub-Riemannian geometry.
method Using nontrivial counterexamples and analysis in Carnot groups.
result The theorem is best possible, with specific gradient restrictions.
Study of Sard problem in Carnot groups using dynamical systems.
problem Sard problem in sub-Riemannian Carnot groups.
method Dynamical-systems approach to study singular curves.
result Positively answer the Sard problem in some Carnot groups.
We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.
Reduced sub-Riemannian time on a specific group structure.
problem Optimizing paths in a sub-Riemannian structure on a Carnot group.
method Proved conjectured cut times, compared with known results, and solved equations in elliptic functions.
result Reduced cut times for sub-Riemannian paths on the Cartan group.
Rectifiability shown for sub-Riemannian manifolds with Carnot tangent structure.
problem Understanding rectifiability in sub-Riemannian manifolds with specific tangent properties.
method Analyzing nilpotentization and embedding properties of sub-Riemannian manifolds into Carnot groups.
result Sub-Riemannian manifolds are countably rectifiable under certain conditions.
The paper constructs optimal sub-Riemannian geodesics in specific Carnot groups.
problem Optimal paths in sub-Riemannian geometry for certain groups.
method Explicit construction of geodesics using symmetries and the Hadamard technique.
result Identification of cut time and cut locus in the constructed geodesics.
Study examines harmonic functions in sub-Riemannian and RCD settings.
problem Characterizing harmonic functions in sub-Riemannian and RCD settings.
method Analyzes weak and strong asymptotically mean value harmonic functions.
result Weakly amv-harmonic functions are equivalent to harmonicity in Carnot groups.
We develope basic geometric quantities and properties of hypersurfaces in Carnot groups.
Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.
problem Classifying metric lines in Engel-type groups.
method Sequence method to study metric lines in jet space.
result Classified metric lines of Engel-type groups $\Eng(n)$.
Study properties of sets with constant normal in Carnot groups.
problem Properties of sets with constant normal in Carnot groups.
method Analysis of subsets with intrinsic constant normal, proving regularity and structural results.
result Every constant-normal set in Carnot groups of step 4 or less is intrinsically rectifiable.
Study geodesics in sub-Riemannian manifolds, resolving open questions.
problem Understanding geodesics in sub-Riemannian geometry, especially those that lose regularity.
method Constructing examples and using a lifting procedure.
result Existence of non-smooth and branching minimizing geodesics in real-analytic sub-Riemannian manifolds and Carnot groups.
In this paper we study the main geometric properties of the Carnot-Carathéodory (abbreviated CC) distance $\dc$ in the setting of k-step sub-Riemannian Carnot groups from many different points of view. An extensive study of the so-called normal CC-geodesics is given. We state and prove some related variational formul…
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.
problem Exploring non-smooth geometries on Lie groups and their applications.
method Study of left-invariant metrics on Lie groups, focusing on nilpotent and Carnot groups.
result Illustrates the role of metric Lie groups, particularly Carnot groups, in various mathematical contexts.
This study classifies metric lines in jet space.
problem Classifying metric lines in jet space.
method Using an intermediate 3D sub-Riemannian space to prove the main theorems.
result Partial results on the classification of metric lines in Jk(R,R). The notion of curvature discussed in this paper is a far going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev, Barilari and Rizzi in arXiv:1306.5318, and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riem…
Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory distance. One of the most striking features of a regular sub-riemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homo…
Let $\GG$ be a sub-Riemannian k-step Carnot group of homogeneous dimension Q. In this paper, we shall prove several geometric inequalities concerning smooth hypersurfaces (i.e. codimension one submanifolds) immersed in $\GG$, endowed with the $\HH$-perimeter measure.
Characterizes GM-groups via sub-Riemannian geometry properties.
problem Characterizing step-two Carnot groups via sub-Riemannian geometry.
method Sub-Riemannian geometric properties, including squared distance, cut locus, optimal synthesis.
result Characterization of GM-groups and exact expression of d(g)2 for classical cut locus. Quantitative estimates for inequalities on sub-Riemannian manifolds.
problem Quantitative estimates for Lp-Poincaré and log-Sobolev inequalities on sub-Riemannian manifolds. method Introducing the Quasi Curvature-Dimension condition and applying it to various sub-Riemannian manifolds.
result Established quantitative estimates independent of the dimension on various sub-Riemannian manifolds.
The author calculates curvatures for homogeneous sub-Riemannian manifolds using specific riggings.
problem Calculating curvatures for homogeneous sub-Riemannian manifolds.
method Using special riggings of invariant completely non-holonomic distributions, the author calculates Solov'ev sectional and Ricci curvatures.
result The method is applicable to contact sub-Riemannian manifolds, sub-Riemannian Carnot groups, and homogeneous sub-Riemannian manifolds with a submetry onto a Riemannian manifold.
Study on extending curves in sub-Riemannian manifolds with compatibility conditions.
problem Validating Whitney extension property for horizontal curves in sub-Riemannian manifolds.
method Analyzing equiregular and singular sub-Riemannian manifolds, using nilpotent approximation and Lusin-like approximation.
result Extension property holds for horizontal curves in sub-Riemannian manifolds with specific conditions.
We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…
Study area and coarea formulas for graphs and submanifolds in Carnot groups.
problem Understanding geometric properties of submanifolds in Carnot groups.
method Developed area and coarea formulas for CH1 intrinsic graphs and submanifolds. result Deduced density properties for Hausdorff measures and coarea formula for Carnot groups.
The paper explores rectifiability in sub-Riemannian geometry, finding a smooth hypersurface with unique properties.
problem Finding a good notion of rectifiability in sub-Riemannian geometry, focusing on smooth hypersurfaces.
method Study of a specific smooth hypersurface in Carnot groups, analyzing its tangent groups and rectifiability properties.
result The existence of a C∞ hypersurface with uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to Michael and Simon, and Allard, independently. Some applications are discussed.
Proves Poincaré lemma on specific sub-Riemannian structures.
problem Solving an open problem on Poincaré lemma for corank 1 sub-Riemannian structures.
method Combines Poincaré lemma on Riemannian manifolds with a path integral formula.
result Necessary and sufficient 'curl-vanishing' conditions for Poincaré lemma.
New insights into integrability and rectifiability in sub-Riemannian geometry.
problem Understanding rectifiability in sub-Riemannian spaces.
method Refined Frobenius Theorem for non-involutive distributions, new metric space class.
result Carnot-Carathéodory spaces are extremal in rectifiability.
Ideal sub-Riemannian manifolds support interpolation inequalities for optimal transport.
problem Optimal transport on sub-Riemannian manifolds.
method Sub-Riemannian Jacobi fields and distortion coefficients.
result Ideal sub-Riemannian manifolds support interpolation inequalities.
Study on distances and neighborhoods in Heisenberg groups, proving regularity and curvature.
problem Analyzing distances and neighborhoods in sub-Riemannian Heisenberg groups.
method Proves H-regularity of Carnot-Carathéodory distance under mild conditions. result Explicit expressions for tubular neighborhoods in terms of principal curvatures and derivatives.
We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the…
New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.
We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…
Optimal quantization of measures on Carnot groups
problem Quantization of probability measures on Carnot groups
method Zador-type asymptotic formula and weak convergence of empirical measures
result Convergence of quantization error and density of absolutely continuous part
The paper studies metrics and geodesics on a quaternionic Heisenberg group.
problem Characterizing geodesics and distances on a quaternionic Heisenberg group.
method Defining and analyzing a sequence of Riemannian metrics, deriving formulas for mean curvature.
result Explicit description of Carnot-Carathéodory distance and spheres.
In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting by Evans-Spruck and Chen-Giga-Goto. We establish two special cases of the comparison p…
Study large deviations in random walks on Lie groups.
problem Large deviations in sub-Riemannian random walks.
method Prove large deviation principle for random walks on stratified Lie groups.
result Proved a large deviation principle with a rate function adapted to sub-Riemannian geometry.
A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step t…
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean C2-smooth surface in the Heisenberg group H away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean C2-smooth curve…
This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.
problem Characterizing semigenerated Carnot groups and their applications to rectifiability.
method Algebraic approach focusing on semigroup generation and Engel-type quotients.
result Complete characterization of semigeneration in Carnot groups of step 3 and sufficient criteria for semigeneration in Carnot groups of arbitrary step.
We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.
In this paper we study heat kernels associated to a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics $σ_\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on G as $\e\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the…