Analyzes properties of transnormal Finsler functions on compact manifolds.
problem Properties of transnormal Finsler functions on compact manifolds.
method Analyzes critical level sets and partition properties of transnormal functions.
result Critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition. Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Study on transnormal functions and their level sets on Finsler manifolds.
problem Understanding transnormal functions and their geometric properties on Finsler manifolds.
method Proving smoothness of focal varieties and regular level sets of transnormal functions.
result Focal varieties of a C2 transnormal function are smooth submanifolds and regular level sets are tubes over these varieties.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
The paper proves properties of Finsler submanifolds and analytic maps.
problem Analyzing properties of Finsler submanifolds and their analytic maps.
method Proving properties of regular fibers of analytic maps and Finsler submersions.
result Regular fibers of an analytic map are equifocal under certain conditions.
Unified geometry for relativity and beyond.
problem Unified geometric framework for relativity and beyond.
method Unified Lorentz-Finsler geometry.
result Unified framework for relativity and beyond.
The paper explores traveling along broken geodesics in Finsler submersions.
problem Analyzing the attainable sets of analytic vector fields in Finsler submersions.
method Investigates the dual leaves and attainable sets of horizontal broken geodesics.
result Proves that in compact Finsler manifolds with positive flag curvature, the attainable sets coincide with orbits.
Busemann G-spaces with Finsler metrics
problem Admitting a differentiable DC atlas with a continuous Finsler metric
method Comparison geometry
result Generalizes previous work on G-spaces with Riemannian curvature bounds
The paper studies curves in Finsler-like spaces and their properties.
problem Investigating properties of curves in asymmetric metric spaces induced by Finsler structures.
method Analyzes three types of absolutely continuous curves in Finsler-like spaces and establishes the Lisini structure theorem.
result Characterizes the nature of absolutely continuous curves in terms of dynamical transference plans.
Researchers developed volume comparison theorems in Finsler spacetimes.
problem Volume comparison in Finsler spacetimes with specific curvature conditions.
method Riccati equation techniques applied to (1+n)-dimensional Lorentz--Finsler manifolds. result Established volume comparison theorems for standard sets in Lorentzian volumes (SCLVs).
In this paper, it is shown that a large set of connections on a suitable sub-bundle of the tangent bundle of a Finsler Manifold can be used to study all the properties of convex neighbourhoods with respect to the Finsler Metric, which are needed to see that any Complete Finsler Space is Geodesically Connected.
Here, the concept of electric capacity on Finsler spaces is introduced and the fundamental conformal invariant property is proved, i.e. the capacity of a compact set on a connected non-compact Finsler manifold is conformal invariant. This work enables mathematicians and theoretical physicists to become more familiar wi…
Study geodesics in Heisenberg groups with sub-Finsler metrics.
problem Characterize infinite geodesics in Heisenberg groups.
method Optimal control theory applied to sub-Finsler metrics.
result Proves infinite geodesics are horizontal lines under specific conditions.
Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality for Finsler manifolds with specific curvature properties.
method Analyzing measured Finsler manifolds with non-negative Ricci curvature and Euclidean volume growth.
result Sharp isoperimetric inequality and rigidity results for the inequality.
Study of perimeter measures in Heisenberg group with sub-Finsler metric.
problem Isoperimetric problem in sub-Finsler Heisenberg group.
method Reduction of Minkowski content to Lebesgue surface area, study of Finsler normed planes, use of CC-geodesics.
result Evidence supports Pansu's conjecture in sub-Finsler case, but with lower isoperimetric ratio.
The paper sets lower bounds for eigenvalues on Finsler manifolds.
problem Understanding the spectrum of Finsler manifolds.
method Analysis of faithful dimension pairs and application of Gromov and Buser types of lower bounds.
result Improved lower bounds for eigenvalues and estimates of multiplicity.
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
Study circumcenters in Finsler unitary groups with optimal convexity bounds.
problem Existence and convexity of circumcenters in Finsler unitary groups.
method Analysis of distance functions and p-Schatten norm on Lie algebra.
result Existence of circumcenters for sets with radius < π/2 in several metrics.
Lower bounds on periodic Finsler billiard trajectories in convex hypersurfaces.
problem Estimating the number of periodic Finsler billiard trajectories.
method Morse and Lusternik-Schnirelmann theories applied to extremal polygons inscribed in a smooth closed hypersurface.
result For prime r≥3, the number of r-periodic Finsler billiard trajectories is not less than (r−1)(d−2)+1. The paper explores families of Finsler metrics and their properties.
problem Understanding symmetrizations and combinations of Finsler metrics.
method Introducing two families of metrics derived from a general Finsler metric.
result Characterization of geodesics, completeness, and Finsler properties of the introduced metrics.
New theory extends classical results to Anosov subgroups.
problem Classical Patterson-Sullivan theory applied to Anosov subgroups.
method Invariant Finsler metrics on symmetric spaces, Gromov pre-metric.
result Equality of Hausdorff dimensions and Finsler critical exponents.
Extends metric to Margulis spacetimes for convex properties.
problem No specific problem stated; extends metric.
method Extends Thurston's asymmetric metric to Margulis spacetimes and proves convex properties.
result Established convex properties of the extended metric.
In this paper, we introduce and investigate a general transformation or change of Finsler metrics, which is referred to as a generalized β-conformal change: L(x,y)⟶L(x,y)=f(eσ(x)L(x,y),β(x,y)). This transformation combines both β-change and conformal change in a general setting. T…
Reconstructing Finsler manifolds from sphere data.
problem Recovering a Finsler manifold from sphere data.
method Solving the geometrical inverse problem locally along geodesics.
result Local reconstruction of Finsler manifolds.
In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space F+(E) of strongly pseudo-convex complex Finsler metrics on E -- a holomorphic vector bundle over a closed Kähler manifold M. This Donaldson type functional is a generalization in the complex…
Study on cut locus of submanifolds in Finsler geometry.
problem Characterizing the cut locus of submanifolds in Finsler manifolds.
method Deformation and characterization of the cut locus, extending previous results.
result Generalization of Klingenberg's lemma for N-geodesic loops in reversible Finsler setting. In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient co…
Using an extension to isometries of the associated Sasaki structure, we establish a Lie transformation group structure for the set of isometries of a pseudo-Finsler conical metric.
The paper extends Sobolev spaces to Finsler manifolds and shows density of smooth functions.
problem Defining and analyzing Sobolev spaces on Finsler manifolds.
method Defining Sobolev spaces for Finsler structures and proving density of smooth functions.
result Smooth functions can approximate weak solutions and functions in Sobolev spaces on Finsler manifolds.
Describes links between Finsler and Lorentz geometries for Riemannian geometers.
problem Understanding the relationship between Finsler and Lorentz geometries.
method Analyzes the Zermelo navigation problem and develops issues related to causality, Finsler elements, and wave propagation.
result Provides a comprehensive understanding of the Lorentzian causality using Finsler elements and the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting.
Study finds first integrals in Finsler metrics with vanishing χ-curvature.
problem Analyzing Finsler metrics with vanishing χ-curvature.
method Proving geodesic invariance of certain structures and deriving first integrals.
result Induces a set of first integrals in Finsler metrics with vanishing χ-curvature.
Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
Paper finds best constants in Hardy inequalities on Finsler metric measure manifolds.
problem Finding best constants in Hardy inequalities on Finsler metric measure manifolds.
method Investigates Hardy inequalities with distance functions in the Finsler setting, considering flag curvature, Ricci curvature, reversibility, and S-curvature.
result Establishes optimal Hardy inequalities on both noncompact and closed Finsler metric measure manifolds.
Determining Finsler manifold structure from boundary distance map and elastic wave measurements.
problem Determine the structure of a compact Finsler manifold from its boundary distance map.
method Construct optimal fiberwise open subset of tangent bundle, use inverse problem in elasticity to measure travel times of waves.
result Finsler function can be uniquely determined from boundary distance map and travel times of waves.
New trigonometric method for convex sets aids control problems.
problem Optimal control problems with two-dimensional control.
method Proposes a new method to describe convex sets.
result Investigates sub-Finsler problems in various groups.
The Weyl principle holds in some Finsler settings despite general failure.
problem Applying the Weyl principle to Finsler manifolds.
method Investigation of the Weyl principle in Finsler geometry.
result A weak form of the Weyl principle persists in certain Finsler settings.
Floer homology proves noncontractible orbits in Finsler manifolds.
problem Existence of noncontractible periodic orbits in Finsler manifolds.
method Floer homology and BPS capacities generalized to Finsler setting.
result Existence of noncontractible periodic orbits for Hamiltonian systems.
Anisotropic connections and parallel transport defined in Finsler spacetimes.
problem Defining and characterizing anisotropic connections and parallel transport in Finsler spacetimes.
method Introducing a new covariant derivative and parallel transport, identifying vertically trivial Finsler connections with anisotropic connections, and characterizing the Levi-Civita-Chern anisotropic connection.
result Characterization of the Levi-Civita-Chern anisotropic connection as the one preserving the length of parallely propagated vectors.
Study of strictly accretive matrices using Finsler geometry.
problem Characterize the set of strictly accretive matrices.
method Introduced Finsler metrics and characterized geodesics and distance.
result Geodesic distance applied to matrix approximation problem.
The paper investigates spectra on Finsler manifolds with properties of eigenvalues and eigenfunctions.
problem Spectral problem on compact Finsler manifolds.
method Definition of spectrum using dimension-like functions and sets in Sobolev space.
result Existence of eigenfunctions and upper bound estimate for eigenvalues.
Study describes periodic controls in step 2 sub-Finsler problems on Carnot groups.
problem Optimal control problems on step 2 Carnot groups with convex control sets.
method Describes Casimirs and symplectic foliations; shows extremal controls are periodic.
result Extremal controls are either constant or periodic.
The paper studies global invertibility of maps on Finsler manifolds.
problem Global invertibility of locally Lipschitz maps on Finsler manifolds.
method Introduces pseudo-Jacobian and studies its relations with local metric properties of the map.
result Conditions for a map to be globally invertible and covering.
The paper explores Finsler-type objects and their variational problems on spacetimes.
problem Generalizing Einstein equations to the Finsler setting.
method Study of the ladder of Finsler-type objects and their variational problems.
result Application of the ladder structure to variational proposals for Finsler spacetimes.
Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.
problem Infinitesimal Hilbertianity of sub-Riemannian manifolds with general measures.
method Embedding metric derivations into square-integrable sections, approximating sub-Finsler distances.
result Sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary Radon measures.
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. We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting.
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.
For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be unique. By means of such connection, an absolute parallelism on U is determined and a…