arXiv research

Solves a PDE for Landsberg surfaces using new Finsler surface insights.

New Finsler metrics derived from pedal curves.

The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.

The paper explores conditions for Finsler surfaces to be Landsbergian and classify surfaces with specific flag curvature conditions.

The study defines Finsler metrics on special surfaces and constructs geodesic currents.

A piecewise flat Finsler metric on a triangulated surface $M$ is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…

This research examines anisotropic conformal transformations of pseudo-Finsler surfaces.

Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on…

Paper proves geodesics are evenly distributed on surfaces.

The study proves surfaces in a specific Heisenberg group must be simple planes.

Paper proves inequality linking capillary surfaces to Finsler geometry.

We study two-dimensional Finsler metrics of constant flag curvature and show that such Finsler metrics that admit a Killing field can be written in a normal form that depends on two arbitrary functions of one variable. Furthermore, we find an approach to calculate these functions for spherically symmetric Finsler surfa…

In this short note, we verify R. Bryant's claim: there does exist the singular Landsberg Finsler surface with a vanishing flag curvature which is not Berwaldian.

New Finsler metrics describe trace function growth rates in convex projective surfaces.

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectr…

Stability result for nearly isometric subspaces and Finsler surfaces.

Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to …

We construct all Finsler metrics on the two-sphere for which geodesics are circles and show that any (reversible) path geometry on a two-dimensional manifold is locally the system of geodesics of a Finsler metric.

We show that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points.

Note refutes examples of Landsberg surfaces with vanishing flag curvature.

We proof that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form.

The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.

Study on special anisotropic conformal changes of conic pseudo-Finsler surfaces.

Paper proves every homogeneous Landsberg surface is either Riemannian or locally Minkowskian.

We show that in dimension 2 every Finsler metric with at least 3-dimensional Lie algebra of projective vector fields is locally projectively equivalent to a Randers metric. We give a short list of such Finsler metrics which is complete up to coordinate change and projective equivalence.

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…

The paper characterizes compatible linear connections on 3D Finsler manifolds.

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas probl…

We consider a general family of curves $Γ$ on a compact oriented Finsler surface $(M,F)$ with boundary $\partial M$. Let $\varphi\in C^{\infty}(M)$ and $ω$ a smooth 1-form on $M$. We show that $\int_{γ(t)}\{\varphi(γ(t))+ω_{γ(t)}(\dotγ(t))\}\,dt=0$ holds for every $γ\inΓ$ whose endpoints belong to $\partial M$, $γ(a)…

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then…

The paper studies invariant functions and their relation to Landsberg surfaces.

The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.

The paper explores families of Finsler metrics and their properties.

The paper studies anisotropic conformal changes in pseudo-Finsler surfaces.

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.

New method uses broken scattering to uniquely identify Finsler manifolds.

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Study of timelike surfaces with time-minimizing rulings in Newtonian and relativistic spacetimes.

The aim of this article is to establish a Toponogov type triangle comparison theorem for Finsler manifolds, in the manner of radial curvature geometry. We consider the situation that the radial flag curvature is bounded below by the radial curvature function of a non-compact surface of revolution, the edge opposite to …

This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.

The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.

New geodesics for surfaces with boundary, extending Thurston's work.

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…

We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent…

The paper characterizes spherically symmetric metrics with scalar curvature.

We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article, as its applications, we prove the finiteness of topological type and a diffeomo…

Given a Finsler space (M,F) on a manifold M, the averaging method associates to Finslerian geometric objects affine geometric objects} living on $M$. In particular, a Riemannian metric is associated to the fundamental tensor $g$ and an affine, torsion free connection is associated to the Chern-Rund connection. As an il…