arXiv research

Smooth Busemann functions found in harmonic Finsler spaces.

The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.

New type of harmonic Finsler manifolds created.

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator), and a $p$-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal $p$-form is harmonic if and only if the horizontal Laplacian vanishes…

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we …

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.

Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.

We study Dirichlet problems for harmonic maps from a Riemannian $m$-manifold $(M,g)$ into a Finsler $n$-manifold $(N, F)$. We assume that the dimension of the source manifold $M$ is less than or equal to 4, and that the finsler structure $F(u,X)$ is given as F(u,X)= \sqrt{h_{ij}(u)X^i X^j + {\cal B}(u,X)}, (u\in N, X \…

The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.

The paper studies Harnack inequalities on Finsler metric measure spaces.

David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain $Ω$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of $Ω$ and refers to a theorem by Busemann and Mayer that…

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami…

Researchers prove constant solutions for a specific Finslerian equation.

The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.

The study introduces new tensors for almost Finsler manifolds and analyzes their properties.

The paper derives inequalities on Finsler manifolds, influenced by their curvatures.

The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.

The study examines Einstein-Finsler spaces using Minkowskian products.

Study on Kähler Finsler manifolds with curvature bounds, proving theorems.

Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at le…

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

The study extends Obata's theorem and classifies Finsler manifolds with transnormal functions.

On the product of two Finsler manifolds M1 M2, we consider the twisted metric F which is construct by using Finsler metrics F1 and F2 on the manifolds M1 and M2, respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study Creducible and semi-C-reducible properties of t…

Most Finsler metrics have infinite-dimensional holonomy groups.

In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}…

The paper proves Hardy inequalities on Finsler manifolds using superharmonicity.

Two special Finsler spaces have been introduced and investigated, namely $R^h$-recurrent Finsler space and consircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor of Cartan connection. The following three results constitute the main object of …

Injectivity of geodesic ray transform on specific Finsler manifolds proven.

Paper studies Minkowskian product of Finsler manifolds and their connections.

The paper extends Montel's theorem to complex Finsler manifolds.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n>2. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler …

The paper shows examples of geodesics switching infinitely often on certain manifolds.

Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.

Study proves conformal vector fields on certain Finsler manifolds are Killing fields.

In this paper, we consider doubly warped product (DWP) Finsler manifolds with some non-Riemannian curvature properties. First, we study Berwald and isotropic mean Berwald DWP-Finsler manifolds. Then we prove that every proper Douglas DWP-Finsler manifold is Riemannian. We show that a proper DWP-manifold is Landsbergian…

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to …

Geodesic walks converge to Brownian motion on Finsler manifolds.

By using a certain second order differential equation, the notion of adapted coordinates on Finsler manifolds is defined and some classifications of complete Finsler manifolds are found. Some examples of Finsler metrics, with positive constant sectional curvature, not necessarily of Randers type nor projectively flat, …

The paper studies Finsler manifolds with a new curvature concept.

In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field l…

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identical…

Established a Hardy inequality on Finsler manifolds.

The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this article, we introduce some important and fundamental topics in global Finsler geometry and discuss the related properties and the relationships in them. In particular, we optimize and improve the various definitions of …

The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.

The aim of this article is to present a comparative review of Riemannian and Finsler geometry. The structures of cut and conjugate loci on Riemannian manifolds have been discussed by many geometers including H. Busemann, M. Berger and W. Klingenberg. The key point in the study of Finsler manifolds is the non-symmetric …

Study on transnormal functions and their level sets on Finsler manifolds.

We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent M…