The paper introduces Carnot coordinates for Carnot manifolds, simplifying nilpotent approximation.
problem Nilpotent approximation of Carnot manifolds.
method Identification of Carnot coordinates as privileged coordinates with specific properties.
result Carnot coordinates provide a precise and effective nilpotent approximation of Carnot manifolds.
The paper defines a differential for Carnot manifold maps and constructs a tangent groupoid.
problem Infinitesimal structure of Carnot manifolds.
method Introduces Carnot differential and constructs tangent groupoid.
result Carnot differential captures the precise deformation of Carnot manifold maps.
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.
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.
This paper clarifies privileged coordinates and nilpotent approximation for Carnot manifolds.
problem Understanding privileged coordinates and nilpotent approximation for Carnot manifolds.
method Systematic account on privileged coordinates and nilpotent approximation of Carnot manifolds.
result Description of all systems of privileged coordinates and algebraic characterization of nilpotent groups.
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.
Explains Gromov's and Rumin's work on Carnot manifolds.
problem Hölder equivalence problem for Carnot manifolds
method PDE techniques and Rumin's complex
result Both methods yield similar conclusions for the H{ö}lder equivalence problem
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.
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.
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.
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.
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…
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…
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. 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.
Characterizes functions in Carnot groups of step 2.
problem Understanding intrinsic Lipschitz functions in Carnot groups.
method Characterization via intrinsic distributional gradients.
result Characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2.
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.
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. 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.
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.
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). 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.
Researchers prove a property for a specific group class, leading to Wasserstein geodesic continuity.
problem Proving a measure contraction property for generalized H-type Carnot groups.
method Analyzing H-type Carnot groups of rank k and dimension n to establish the MCP(K,N) condition. result Generalized H-type Carnot groups satisfy MCP(K,N) with K≤0 and N≥k+3(n−k), matching geodesic dimension. 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.
Step 2 Carnot groups can approximate horizontal curves without error.
problem Approximating horizontal curves in step 2 Carnot groups.
method Verification of Lusin approximation for free Carnot groups of step 2 and preservation by homomorphisms.
result All step 2 Carnot groups admit Lusin approximation for horizontal curves.
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.
Carnot groups are studied as special graded groups and homogeneous metric spaces.
problem Understanding the structure and properties of Carnot groups.
method Presentation of basic theory, discussion of isometries, and classification as special cases of graded groups and homogeneous metric spaces.
result Regularity of isometries in Carnot-Caratheodory spaces and nilpotent metric Lie groups.
Proves codimension of abnormal set in step 2 Carnot groups is at least 3.
problem Determining the codimension of abnormal set in Carnot groups.
method Analyzes endpoint map for specific Carnot groups of step 2.
result Codimension of abnormal set is at least 3 for all step 2 Carnot groups up to dimension 7.
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.
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…
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.
Study intrinsic regular surfaces in Carnot groups, generalizing results from Heisenberg groups.
problem Equivalence of definitions of intrinsic regular surfaces in Carnot groups.
method Generalize results from Heisenberg groups to Carnot groups.
result Equivalence of definitions of intrinsic regular surfaces in Carnot groups.
New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
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.
Research examines cohomology of Carnot groups, finding conditions for vanishing and non-vanishing.
problem Understanding cohomology of Carnot groups under different conditions.
method Analyzes simplicial \ell q,p cohomology of Carnot groups G, considering (p, q) and weight gaps.
result Shows conditions for vanishing and non-vanishing of cohomology.
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 …
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.
Study curvature in Carnot groups with rank-two distributions.
problem Understanding curvature in specific geometric structures.
method Generalized sectional curvature for Carnot groups with rank-two distributions.
result Curvature depends on the Engel part in certain 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.
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.
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.