This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (…
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.
Study hypoelliptic operators on Carnot manifolds, extending index theory results.
problem Index theory of hypoelliptic operators on Carnot manifolds.
method Operator K-theory and geometric K-homology.
result Compute Fredholm index of hypoelliptic operators on Carnot manifolds.
Study on mappings in Carnot groups, proving rigidity results.
problem Understanding mappings in Carnot groups and proving rigidity.
method Structural results for Sobolev mappings, proving rigidity or regularity.
result Establishes partial rigidity and partial regularity theorems.
This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents…
In this note we show how results in \cite{BaudoinBonnefont2016, BaudoinGarofalo2013, CoulhonJiangKoskelaSikora2017} yield the Cheng-Yau estimate on two classes of sub-Riemannian manifolds: Carnot groups and sub-Riemannian manifolds satisfying a generalized curvature-dimension inequality.
Extends scaling maps theory to manifolds with boundary.
problem Quantitative study of Carnot-Carathéodory balls on manifolds with boundary.
method Introduction of scaling maps adapted to Carnot-Carathéodory balls and Hörmander vector fields on manifolds with boundary.
result First paper in a series studying maximally subelliptic boundary value problems.
In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies …
We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.
The paper explores the index theory of sub-Laplacians on higher nilpotent Carnot manifolds.
problem Characterizing the index theory of sub-Laplacians on higher nilpotent Carnot manifolds.
method Analyzes the structure of hypoelliptic sub-Laplacian type operators and provides examples where the index theory is trivial.
result Provides examples where the index theory of sub-Laplacians is trivial in higher degrees of nilpotency.
Sobolev mappings preserve the Rumin complex on contact manifolds.
problem Preserving the Rumin complex under Sobolev mappings on contact manifolds.
method Using the Pullback Theorem, Pansu pullback is shown to induce chain mappings between Rumin complexes and de Rham complexes.
result The Rumin flat complex is bilipschitz invariant under Sobolev mappings between contact manifolds.
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.
The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…
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 $\Eng(n)$.
The paper examines convergence of distances in Lipschitz structures on manifolds.
problem Convergence of distances in Lipschitz vector fields and norms on manifolds.
method Analysis of convergence of distances associated to converging structures of Lipschitz vector fields and norms.
result Under mild controllability assumption, distances converge locally uniformly to the limit Carnot-Carathéodory distance.
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…
We show that if M is a sub-Riemannian manifold and N is a Carnot group such that the nilpotentization of M at almost every point is isomorphic to N, then there are subsets of N of positive measure that embed into M by bilipschitz maps. Furthermore, M is countably N--rectifiable, i.e., all of M except …
Smooth C1 contact maps are always smooth in rigid Carnot groups.
problem Smoothness of C1 contact maps in rigid Carnot groups. method Analyzing C∞-rigid Carnot groups to show C1-contact maps are smooth. result Smooth C1 contact maps are always smooth in rigid Carnot groups. The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--C…
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.
Improved Sobolev mappings in Carnot groups with weaker assumptions.
problem Improving Sobolev mappings in Carnot groups with weaker conditions.
method Using Buser-Karcher center-of-mass and polynomial expressions in moments.
result Rigidity and structural results hold under weaker Sobolev exponents.
Rigidity theorem for flag manifolds in various dimensions.
problem Rigidity of flag manifolds under certain mappings.
method Rigidity theorem derived from quasiconformal homeomorphisms and Sobolev mappings.
result Quasiconformal homeomorphisms and Sobolev mappings are rigid for flag manifolds in dimensions n≥4. The paper shows examples of geodesics switching infinitely often on certain manifolds.
problem Understanding geodesics with infinitely many switches on Finsler and sub-Finsler manifolds.
method Provided examples and explicit structures on Carnot groups, presented a sufficient condition for chattering.
result Geodesics on certain manifolds can exhibit a countable number of switches in arbitrarily small time intervals.
Curves in Carnot groups avoid compact sets, growing at least t1/s.
problem Existence of periodic normal geodesics in subFinsler Carnot groups.
method Analysis of curves satisfying Pontryagin Maximum Principle.
result Normal curves in subFinsler Carnot groups leave every compact set.
New bounds on geodesic dimension and curvature exponent in Carnot groups.
problem Characterizing geodesic dimension and curvature exponent in Carnot groups.
method Characterization and lower bound calculation for geodesic dimension and curvature exponent.
result Found an example where curvature exponent is greater than geodesic dimension.
We prove that H-type Carnot groups of rank k and dimension n satisfy the MCP(K,N) if and only if K≤0 and N≥k+3(n−k). The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in…
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
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). Compact currents and charges in Carnot groups proved.
problem Compactness of normal currents in Carnot groups.
method Dual compactness argument for Rumin forms using pseudo-differential calculus.
result Compactness of normal currents in Carnot groups in flat topology.
ODE trajectories become abnormal curves in Carnot groups.
problem Understanding abnormal curves in Carnot groups.
method Explicit construction of covectors for abnormal curves.
result Polynomial ODE trajectories lift to abnormal curves in Carnot groups.
Study on mappings between nonrigid Carnot groups, proving quasisymmetric rigidity.
problem Quasisymmetric homeomorphisms in nonrigid Carnot groups.
method Use pullback theorem from previous work to show reducibility and rigidity.
result Quasisymmetric homeomorphisms are reducible in nonrigid Carnot groups, except for specific cases.
We show that isometries between open sets of Carnot groups are affine. This result generalizes a result of Hamenstadt. Our proof does not rely on her proof. In addition, we study global isometries of general homogeneous manifolds equipped with left-invariant subFinsler distances. We show that each isometry is determine…
This paper studies rectifiability in Carnot groups and proves geometric area formulas.
problem The study of rectifiability in Carnot groups and related geometric properties.
method Analysis of rectifiable measures in Carnot groups, geometric area formulas, and rectifiability of geodesic spheres.
result Geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in Carnot groups.
Commutes Pansu pullback with spectral complexes in Carnot groups.
problem Understanding the relationship between Pansu pullback and spectral complexes in Carnot groups.
method Proving commutativity between Pansu pullback and differentials in spectral complexes.
result Commutes Pansu pullback with spectral complexes in Carnot groups.
The paper explores the Rumin complex and spectral sequence on Carnot groups.
problem Understanding the relationship between Rumin complex and spectral sequence on Carnot groups.
method Investigates the Rumin complex and spectral sequence on Carnot groups, focusing on the filtration by homogeneous weights.
result Provides a detailed insight into the relationship between the Rumin complex and the spectral sequence on Carnot groups.
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known …
Study shows a specific Carnot group violates a curvature exponent bound.
problem Understanding the curvature exponent in step-two Carnot groups.
method Examined convergence of Lie algebra structure constants.
result Found a Carnot group where curvature exponent bound is violated.
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.
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
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.
We characterize the rigidity of Carnot groups in the class of C2 contact maps in terms of complex characteristics. Furthermore, we obtain a Liouville type theorem for Carnot groups which states that 1-quasiconformal maps form finite dimensional Lie groups.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
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.
Extends potential theory to Carnot groups, estimating Hausdorff dimension.
problem Estimating Hausdorff dimension of polar sets in Carnot groups.
method Geometric completeness and Riesz potential inequalities in Carnot groups.
result Developed applications in CR geometry and quaternionic CR geometry.
Paper proves equivalence of derivatives for maps between Carnot groups.
problem Maps between Carnot groups and their derivatives.
method Elementary proof using Euclidean arguments and mean value estimates.
result Maps preserving horizontal curves are continuously Pansu differentiable.
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
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