New metrics for Anosov representations defined from Thurston's asymmetric metrics.
problem Defining metrics for Anosov representations.
method Generalizing Thurston's asymmetric metric to Anosov representations.
result Provides a (possibly asymmetric) Finsler distance in some cases.
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.
We highlight several analogies between the Finsler (infinitesimal) properties of Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space. Thurston defined his asymmetric metric in analogy with Teichmüllers' metric, as a solution to an extremal problem, which consists, in the case of the asymmetric me…
Paper proves rigidity theorems for geodesically reversible Finsler metrics.
problem Understanding geodesically reversible Finsler metrics in closed manifolds.
method Applied theory of volumes and areas on Finsler spaces to establish rigidity theorems.
result Partial explanation of the scarcity of geodesically reversible Finsler metrics in closed manifolds.
This paper studies gradient flows in asymmetric metric spaces and proves existence results.
problem Investigating gradient flows in asymmetric metric spaces.
method Discrete approximation and natural convexity assumption on potential function.
result Existence of curves of maximal slope in asymmetric metric spaces.
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective n-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…
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.
The paper describes geometric properties of Teichmüller space metrics.
problem Analyzing the geometry of Teichmüller space with weak Finsler metrics.
method Geometric description of unit spheres in weak Finsler metrics.
result Introduced a family of weak Finsler metrics interpolating between Thurston's metric and Teichmüller metric.
On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting met…
Characterizes isometries between non-reversible Finsler manifolds.
problem Understanding isometries in non-reversible Finsler manifolds.
method Generalization of Myers-Nakai Theorem for Riemannian manifolds, modification of function spaces to accommodate asymmetric structure.
result Functional characterization of isometries between non-reversible Finsler manifolds.
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.
Study bounds on curvature for special Finsler metrics.
problem Curvature and topological properties of ∞-Einstein Finsler metrics. method Construct special metrics, analyze equivalence, impose curvature bounds.
result Establish bounds for curvature and distortion on ∞-Einstein Finsler manifolds. Warped product construction for Finsler manifolds with curvature conditions.
problem Constructing new Finslerian manifolds with specific curvature properties.
method Extending the warped product concept to Finsler metrics, particularly (α,β)-metrics. result Demonstrated the feasibility of constructing new Finslerian manifolds with prescribed curvature conditions.
The paper defines a metric on Euclidean triangles and polygons, proving properties and completeness.
problem Defining and analyzing a metric space for Euclidean triangles and polygons.
method Introducing and proving properties of a metric on marked Euclidean triangles, extending to polygons and triangulated surfaces.
result The metric is Finsler and complete, providing formulas for its infinitesimal structure.
Study on earthquake metric on Teichmüller space, proving properties and new completions.
problem Understanding the earthquake metric on Teichmüller space.
method Proofs of properties, new completions, and interpretation of the metric.
result Coincidence of various completions for the earthquake metric.
New insights into currents of Hitchin representations with combinatorial restrictions.
problem Understanding currents associated with Hitchin representations.
method Defining dual spaces and analyzing combinatorial restrictions on self-intersection.
result Dual spaces of discrete boundary currents are polyhedral complexes with dimension at most n-1.
New Finsler metrics describe trace function growth rates in convex projective surfaces.
problem Understanding growth rates of trace functions in convex projective surfaces.
method Introduced new Finsler metrics and showed their convergence to describe trace function growth.
result Logarithms of trace functions are approximated by lengths in a Finsler metric defined by cubic differential.
New friction model for geometric locomotion systems.
problem Modeling asymmetric friction in locomotion systems.
method Introducing asymmetric friction into geometric locomotion models using Finsler metrics.
result Generalized motility map for systems with asymmetric friction.
New geodesics for surfaces with boundary, extending Thurston's work.
problem Extending Thurston's Lipschitz distance to surfaces with boundaries.
method Constructing geodesics in the arc distance metric space.
result Teichmueller space of surfaces with boundaries is a geodesic metric space.
Maps between acute triangles with minimal stretch found and studied.
problem Finding the minimal stretch between acute triangles.
method Formula for the smallest Lipschitz constant and analysis of the metric space.
result Metric space of pairs of acute triangles with fixed area is Finsler and geodesics determined.
We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential Ψon Outer space such that when the Lipschitz norm is corrected by the derivative of Ψ, the resulting norm is quasisymmetric. As an application, we give new …
Study extremals on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle.
problem Finding extremals on Lie groups with asymmetric polyhedral Finsler structures.
method Using Pontryagin's Maximal Principle and control systems of Euler-Arnold type to find extremals on the cotangent bundle of the group.
result Uniqueness of the control u(t) can be studied through the asymptotic curvature of the vertical part of the Pontryagin extremal. Study smooths Finsler structures on Lie groups, proving extremal convergence.
problem Smooth left-invariant strongly convex C0-Finsler structures on Lie groups. method Introduce mollifier smoothing, study extremals using Pontryagin maximum principle.
result Pontryagin extremals on smoothed Finsler structures converge uniformly to those on original structure.
Generalizes Thurston's asymmetric metric to flat metrics.
problem Defining an asymmetric metric on flat metrics.
method Defined an asymmetric metric on the space of unit-area flat metrics.
result Discussed two different topologies from the asymmetry.
Characterizes complex Finsler metrics and their properties.
problem Characterize complex Finsler metrics and their geometric properties.
method Defined the canonical connection and investigated holomorphic sectional curvature tensors and Ricci curvatures.
result Characterizes balanced complex Finsler metrics and provides sufficient and necessary conditions.
Smooths C0-Finsler structures using mollifiers.
problem Constructing smooth approximations of discontinuous Finsler structures.
method Convolution of C0-Finsler structure with standard mollifier to create C∞-Finsler structures. result Uniform convergence of Finsler structures and their geometric properties to the original structure.
Study on special Finsler metrics with conditions for Riemannian and isotropic properties.
problem Characterizing Finsler metrics with specific geometric properties.
method Analyzing conditions for Riemannian and isotropic properties of AR-Finsler metrics.
result Conditions for isotropic S-curvature and mean Landsberg curvature leading to vanishing curvature. Introduces new geodesic fields for Finsler manifolds.
problem Finding optimal paths in Finsler manifolds.
method Introduced Pontryagin type C0-Finsler structures and defined geodesic fields. result Extended geodesic field E provides more natural geodesics. 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.
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.
Paper studies Landsberg curvature of a specific Finsler metric.
problem Analyzing Landsberg curvature of a particular Finsler metric.
method Derived Landsberg curvature and mean Landsberg curvature of the twisted product Finsler metric.
result Necessary and sufficient conditions for the metric to be Landsberg or weakly Landsberg are established.
New Finsler metrics constructed from GDW-metrics.
problem Exploring new Finsler metrics within the GDW-metric class. method Constructing new sub-classes of GDW-metrics. result Presented illustrative examples of new Finsler metrics.
Investigate AR-Finsler metrics for local dual flatness and projective flatness.
problem Locally dually and projectively flat AR-Finsler metrics
method Derive necessary and sufficient conditions and a compatibility relation.
result Establish a rigidity result for AR-Finsler metrics.
Study Ricci curvature of homogeneous Finsler spaces with specific metrics.
problem Curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Derived explicit formulae for Ricci curvature and found conditions for vanishing S-curvature. result Spaces with vanishing S-curvature and negative Ricci curvature are Riemannian. New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.
Study cylindrical symmetric Finsler metrics with vanishing Douglas curvature.
problem Understanding Finsler metrics with specific curvature properties.
method Analyzing cylindrically symmetric Finsler metrics and solving differential equations.
result Differential equations for cylindrically symmetric Finsler metrics with vanishing Douglas curvature.
This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.
problem Understanding the properties of Finsler metrics and their projective invariants.
method Developed and examined weakly-Weyl and generalized weakly-Weyl Finsler metrics.
result Equivalence of weakly-Weyl and W-quadratic spherically symmetric Finsler metrics. We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we …
Study of classification in asymmetric quasi-metric spaces.
problem Classification in asymmetric quasi-metric spaces.
method Sample compression and nearest neighbor algorithm.
result Algorithm has favorable statistical properties.
Study flag curvature in homogeneous Finsler spaces with a specific metric.
problem Analyzing flag curvature in homogeneous Finsler spaces with a generalized m-Kropina metric. method Provided explicit formula for flag curvature, showed equivalence of definitions, and studied curvature of naturally reductive spaces.
result Equivalence of two definitions of naturally reductive homogeneous Finsler spaces for the generalized m-Kropina metric. The paper examines Finsler metrics under Ricci flow and finds they are Einstein.
problem Understanding Finsler metrics under Ricci flow.
method Investigated C3-like Finsler metrics and proved they are Einstein.
result C3-like Finsler metrics are Einstein under Ricci flow.
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
Study curvature of complex Finsler metrics on Lie groups.
problem Explicitly calculate and characterize curvature of complex Finsler metrics.
method Analyzes left-invariant complex Finsler metrics on Lie groups using their Lie algebra.
result Characterizes conditions for a Finsler metric to be Kähler or weakly Kähler.
New Finsler metrics derived from pedal curves.
problem Constructing new Finsler metrics.
method Using pedal curves or surfaces of other curves/surfaces.
result Generalization of slope metric.
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…