Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.
The tangent space is constructed in sub-Finsler geometry, leading to the failure of the CD condition in 3D-contact manifolds.
problem The failure of the CD condition in sub-Finsler geometry.
method Construction of the tangent space in the measured Gromov-Hausdorff sense, application of nilpotent approximation.
result The CD condition fails in 3D-contact sub-Finsler manifolds.
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.
Researchers find optimal paths on a specific geometric group.
problem Finding optimal paths on a Cartan group with a sub-Finsler quasimetric.
method Using the Pontryagin Maximum Principle in coordinates of the first kind.
result They found extremals for arbitrary left-invariant sub-Finsler quasimetrics.
In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…
The paper examines curvature-dimension bounds on sub-Finsler Heisenberg groups.
problem Investigating synthetic curvature-dimension bounds in sub-Finsler Heisenberg groups.
method Study of measure contraction property (MCP) and curvature-dimension condition (CD).
result Sub-Finsler Heisenberg groups do not satisfy MCP or CD for any parameters.
We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [5] and ha…
The study proves surfaces in a specific Heisenberg group must be simple planes.
problem Characterizing surfaces in a sub-Finsler Heisenberg group.
method Analyzes (X,Y)-Lipschitz surfaces in H1 with a sub-Finsler structure. result Complete, oriented, stable (X,Y)-Lipschitz surfaces are vertical planes. The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.
problem The failure of the curvature-dimension condition in sub-Finsler geometry.
method Non-trivial adaptation of Juillet's work, introduction of new tools and ideas.
result The CD(K,N) condition does not hold in sub-Finsler geometry for various norms and measures. Study of billiards in sub-Finsler geometry, including unusual orbits.
problem Exploring billiard dynamics in sub-Finsler spaces.
method Symplectic and variational approaches, control theory.
result Unusual orbits like gliding and creeping orbits exist.
Researchers found abnormal extremals on specific Lie groups.
problem Identifying abnormal extremals on four-dimensional Lie groups.
method Using left-invariant sub-Finsler quasimetrics and seminorms on Lie algebra.
result Established a criterion for strict abnormality of extremals.
Study on curvature bounds and geodesic dimension in sub-Finsler Heisenberg groups.
problem Investigate synthetic curvature-dimension bounds in sub-Finsler geometry.
method Examine measure contraction property and geodesic dimension on Heisenberg groups with ℓp-sub-Finsler norms. result For p∈(2,∞], ℓp-Heisenberg group fails to satisfy any measure contraction property. For p∈(1,2), it satisfies MCP(K,N) under specific conditions. Study finds abnormal paths on specific Lie groups using algebraic structures.
problem Identifying abnormal extremals on Lie groups with quasimetrics.
method Analyzing Lie algebras and seminorms to determine abnormal extremals.
result Established criterion for strong abnormality of extremals.
We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.
Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
Study on curvature equation in Heisenberg group with convex boundary.
problem Existence of solutions to prescribed mean curvature equation in sub-Finsler Heisenberg group.
method Finsler approximation scheme to prove existence of Lipschitz solutions.
result Existence of a Lipschitz solution for the Dirichlet problem.
Study on nonholonomic mechanics and sub-Finsler geometry.
problem Understanding nonholonomic mechanical systems and their geometric properties.
method Variational approach, sub-Finsler manifolds, nonholonomic sub-Finslerian structure.
result Existence and properties of extremals in nonholonomic mechanics.
Researchers found explicit solutions to a complex equation in advanced geometry.
problem Critical Yamabe type equation in sub-Finsler geometry.
method Computed a two-parameter family of explicit positive solutions.
result Explicit solutions to a critical equation in sub-Finsler geometry.
Optimally regularizes boundaries in the Heisenberg group with prescribed curvature.
problem Optimizing boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group.
method Analyzes critical sets of the prescribed mean curvature functional in the Heisenberg group.
result Characteristic curves of critical sets are C2-regular, optimal in the Heisenberg group. The paper derives explicit geodesic equations for a specific type of group structure.
problem Finding geodesics in left-invariant sub-Finsler problems on Heisenberg groups.
method Using convex trigonometry and generalizations of spherical coordinates.
result Explicit formulae for geodesics in Heisenberg groups are derived.
The authors found extremals of arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank two. They use for this the Pontryagin Maximum Principle for the corresponding time-optimal problem in coordinates of the first kind. The obtained results are applied to the case of left-invari…
The study solves the isoperimetric problem for Heisenberg group norms.
problem Solving the isoperimetric problem for anisotropic norms in the Heisenberg group.
method Representation formula for perimeter, foliation property, differential equation characterization, approximation procedure.
result Characterization of isoperimetric sets as sub-Finsler analogues of Pansu's bubbles.
In this paper we study a sub-Finsler geometric problem on the free-nilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the ℓ∞ norm on its first layer. We adopt the point of view of time-optimal control theory…
We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all linear-in-momenta Casimirs on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation o…
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on L…
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all we…
Study proves existence of regions minimizing perimeter in specific geometric structures.
problem Existence of isoperimetric regions in sub-Finsler nilpotent groups.
method Analyzes nilpotent Lie groups with a bracket-generating distribution and asymmetric norms.
result Proves existence of minimizers of perimeter under volume constraint.
This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface ar…
A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions sin and cos. Apparently, this method may be very useful for explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Fin…
Proves rectifiability for specific metric spaces with unique tangents.
problem Rectifiability of CD(K,N) and MCP(K,N) spaces with unique tangents. method Failure of CD condition in sub-Finsler Carnot groups, new result on MCP spaces, recent breakthrough by Bate. result Proves rectifiability for CD(K,N) and MCP(K,N) spaces under specific conditions. This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler ℓ∞ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that…
The paper solves optimal control problems for various convex sets using convex trigonometry.
problem Optimal control problems with 2D convex compact sets.
method Using convex trigonometry to derive extremals for various problems.
result Geodesics in multiple sub-Finsler problems are derived.
Examples of area-minimizing graphs with low regularity in a specific group.
problem Finding area-minimizing graphs with low regularity in a sub-Finsler Heisenberg group.
method Providing examples of entire area-minimizing horizontal graphs with prescribed singular sets.
result Examples of area-minimizing graphs that are locally Lipschitz but not necessarily smoother.
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…
The Hopf-Rinow theorem is extended to sub-Finslerian geometry.
problem Extending the Hopf-Rinow theorem to sub-Finslerian manifolds.
method Investigation of sub-Finslerian bundle, exponential map, and Legendre transformation.
result Established a relation between completeness, geodesic completeness, and compactness in sub-Finslerian geometry.
The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is p…
The paper studies metrics on hyperkähler manifolds using sub-twistor constraints.
problem Understanding metrics on hyperkähler manifolds with non-holonomic constraints.
method Introducing and analyzing sub-conic metrics, showing they are sub-Finsler.
result Explicit description of Finsler norms for sub-twistor metrics.
The paper studies distributions and controllability in quantum mechanical systems.
problem Controlling quantum mechanical systems and their evolution.
method Analysis of distributions, controllability, and geodesics on sub-Finsler manifolds.
result Proves the Lie group decomposition and geodesics equivalence for quantum system steering.
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is…
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.
Introduce sub-Randers metrics by adding a one-form to a sub-Riemannian metric
problem Define a new class of sub-Finsler metrics
method Derive equations for sub-Randers normal geodesics
result Prove a Hopf-Rinow type theorem for sub-Randers manifolds
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…
Study on shapes in Heisenberg group with convex body norms.
problem Characterize shapes in the Heisenberg group H1 induced by convex bodies. method Compute perimeter variation, define mean curvature, and analyze foliations.
result Existence of constant mean curvature spheres in the Heisenberg group.