The isoperimetric problem in the 2-dimensional Finsler space forms with k = 0. I

Finsler几何的半群方法：Bakry-Ledoux等周不等式

A semigroup approach to Finsler geometry: Bakry-Ledoux's isoperimetric inequality

本文在Finsler流形上建立Bakry半群方法，由$L^1$-梯度估计与曲率下界推出Bakry-Ledoux高斯等周不等式，并推广至非可逆情形。

The Isoperimetric Problem In Randers Planes

In this paper, the isoperimetric problem in Randers planes, $(\mathbb{R}^2,F=α+β)$, which are slight deformation of the Euclidean plane $(\mathbb{R}^2,α)$ by suitable one forms $β$, have been studied. We prove that the circles centred at the origin achieves the local maximum area of the isoperimetric problem with respect to well known volume forms in Finsler geometry.

The Isoperimetric Problem in Randers Poincaré Disc

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this article, we show that Randers Poincaré disc satisfies the isoperimetric equality with respect to different volume forms. These metrics do not necessarily have constant (negative) flag curvature. Thus we show that the Osserman's result \cite{RO} in the Riemannian case can not be extended to the corresponding Finslerian case.

A rigidity result of spectral gap on Finsler manifolds and its application

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a diffeomorphic splitting (or isometric splitting in the particular class of Berwald spaces). This splitting phenomenon is comparable to the Cheeger-Gromoll type splitting theorem by Ohta. We also obtain the rigidity results of logarithmic Sobolev and Bakry-Ledoux isoperimetric inequalities via needle decomposition as corollaries.

The isoperimetric problem for regular and crystalline norms in $\mathbb H^1$

We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $ϕ$ on the horizontal distribution. We first prove a representation formula for the $ϕ$-perimeter of regular sets and, assuming some regularity on $ϕ$ and on its dual norm $ϕ^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $ϕ$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal…

On minimal immersions in Finsler spaces

We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we establish the existence of smooth Finsler-minimal immersions spanning given extreme or graphlike boundary contours.

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.

Plateau's problem in Finsler 3-space

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 problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.

Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our r…

Majorization by Hemispheres & Quadratic Isoperimetric Constants

Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $γ:S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the hemisphere $f:H^2\rightarrow X$. This implies that $X$ satisfies a quadratic isoperimetric inequality (for curves) with constant $\frac{1}{2π}$. We discuss how this fact controls the regularity of minimal discs in Finsler manifolds when applied to the work of Alexander Lytchak and Stefan Wenger.

Horizontal Forms of Chern Type on Complex Finsler Bundles

The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied.

Isoparametric hypersurfaces in Finsler space forms

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.

Homogeneous Finsler spaces with some special $(α, β)$-metrics

In this paper, first we prove the existence of invariant vector field on a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce an explicit formula for the the $S$-curvature of homogeneous Finsler space with these metrics. Using this formula, we further derive the formula for mean Berwald curvature of the homogeneous Finsler space with the above mentioned metrics.

Killing frames and S-curvature of homogeneous Finsler spaces

In this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we give an explicit formula which coincides with the previous formula obtained by the second author using other methods.

On Horizontal Recurrent Finsler Connections

In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($π$)1-form $A$, there exists a unique horizontally recurrent Finsler connection whose $h$-recurrence form is $A$. This result generalizes the existence and uniqueness theorem of Cartan connection. We then study some properties of a special kind of horizontally recurrent Finsler connection, which we call special HRF-connection.

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $π:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler-Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Fi…

Finsler空间形式上的等参函数

Isoparametric functions on Finsler space forms

本文在Finsler空间形式$(N(c),F)$上证明在一定条件下超正则函数是等参函数，推广了Riemann情形的定理B；并在前向Finsler空间形式$(\overrightarrow{N}(c),F)$上讨论脐超曲面与等参超曲面的关系，利用距离函数构造全局和局部等参函数。

Spherically symmetric Finsler metrics in R^n

In this paper, we give the general form of spherically symmetric Finsler metrics in $R^n$ and surprisedly find that many well-known Finsler metrics belong to this class. Then we explicitly express projective metrics of this type. The necessary and sufficient conditions that projective Finsler metrics with spherical symmetry have constant flag curvature are also obtained.

On the Ricci curvature of homogeneous Finsler spaces with $(α,β)$-metrics

The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In this paper, we consider homogeneous Finsler spaces with square metric and Randers change of square metric. First, we derive the explicit formulae for Ricci curvature of homogeneous Finsler spaces with these metrics. Next, we find a necessary and sufficient condition under which a homogeneous Finsler space with either of these metrics is of vanishing $S$-curvature. The formulae for Ricci curvature of homogeneous Finsler spaces with square metric an…

On $S$-Curvature of Homogeneous Finsler spaces with $(α, β)$-metrics

The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with square $(α, β)$-metric and Randers changed square $(α, β)$-metric is proved. Further, an explicit formula for $S$-curvature of these $(α, β)$-metrics is established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(α, β)$-metrics is calculated.

Jellett-Minkowski's formula revisited. Isoperimetric inequalities for submanifolds in an ambient manifold with bounded curvature

In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with st…

On quasi-Eienstein Finsler spaces

The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces. Quasi-Einstein metrics serve also as solution to the Ricci flow equation. Here, the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined. In compact case, it is proved that the quasi-Einstein metrics are solution to the Finslerian Ricci flow and conversely, certain form of solutions to the Finslerian Ricci flow are quasi-Einstein Finsler metrics.

On a class of Einstein Finsler metrics

In this paper, we study a class of Finsler metrics called general (α,β)-metrics, which are defined by a Riemannian metric and an 1-form. We construct some general (α,β)-metrics with constant Ricci curvature.

A de Rham theorem with respect to the Liouville foliation on TM^0, for a Finsler manifold M

On the slit tangent manifold of a Finsler manifold M are given the vertical and the Liouville foliations. In this paper we define some new types of vertical forms with respect to the Liouville foliation on TM^0. We define a cohomology group of TM^0 using these new forms. We prove a de Rham type theorem.

On the flag curvature of a homogeneous Finsler sapce with generalized $m$-Kropina metric

In this paper, first, we give an explicit formula for the flag curvature of a homogeneous Finsler space with generalized $m$-Kropina metric. Then, we show that, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space are equivalent for afore said metric. Finally, we study the flag curvature of naturally reductive homogeneous Finsler spaces with generalized $m$-Kropina metric.

Even dimensional homogeneous Finsler spaces with positive flag curvature

In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out all the coset spaces admitting invariant non-Riemannian Finsler metrics with positive flag curvature.

On the flag curvature of homogeneous Finsler space with some special $(α, β)$-metrics

In this paper, first we derive an explicit formula for the flag curvature of a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce it for naturally reductive homogeneous Finsler space with the above mentioned metrics.

Towards the classification of odd dimensional homogeneous reversible Finsler spaces with positive flag curvature

In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional positively curved Riemannian homogeneous spaces can be generalized to reversible Finsler spaces.

Frobenius integrability and Finsler metrizability for $2$-dimensional sprays

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous $1$-form from the annihilator of the Berwald distribution. We discuss both the degenerate and non-degenerate cases using the fact that the regularity of the Finsler function is encoded into a regularity condition of a $2$-form, canonically associate…

特殊芬斯勒流形上的平行1-形式

Parallel one forms on special Finsler manifolds

本文研究特殊芬斯勒流形上平行1-形式的存在性，推导了Landsberg流形相关曲率秩的结论，证明了几类特殊芬斯勒度量均不容许平行1-形式。

A variational problem for curves on Riemann-Finsler surfaces

We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem.

Spherically Symmetric Finsler Metrics With Constant Ricci And Flag Curvature

Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.

On Finslerized Absolute Parallelism spaces

The aim of the present paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism space (GAP-space). The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized Parallelizable space, combines within its geometric structure the simplicity of GAP-geometry and the richness of Finsler geometry, hence is potentially more suitable for applications and especially for describing physical phenomena. A study of the geometry of the two structures and…

Curvatures on homogeneous Finsler spaces

The main aim of this article is to calculate explicit formula for S-curvature in homogeneous generalized m-Kropina metric. Further, we also deduce mean Berwald curvature for homogeneous generalized m-Kropina metric from S-curvature.

Finsler 几何上的调和向量场与 Hodge Laplacian 算子

Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry

本文在 Finsler 流形上定义了水平微分、散度与 $p$-调和形式，证明了 Hodge 型定理，提出了调和向量场的新定义，并导出了基于调和 Ricci 标量的 Bochner-Yano 型分类定理。

An observation on Asanov's Unicorn metrics

Finsleroid-Finsler metrics form an important class of singular (y-local) Finslerian metrics. They were introduced by G. S. Asanov in 2006. As a special case Asanov produced examples of Landsberg spaces of dimension at least three that are not of Berwald type. These are called Unicorns [5]. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day. In this paper we prove that Asanov's Unicorns belong to the class of generalized Berwald manifolds. More precisely we prove the following theorems: a Finsleroid-Finsler space is a generaliz…

一些同质Finsler流形中的两步同质测地线

Two-step homogeneous geodesics in some homogeneous Finsler manifolds

本文探讨了同质Finsler空间中两步Finsler测地轨道空间的条件及示例，扩展了同质Riemannian空间中同质测地线的概念。

Finsleroid--Finsler Space with Berwald and Landsberg Conditions

We formulate the notion of the Finsleroid--Finsler space, including the positive--definite as well as indefinite cases. The associated concepts of angle, scalar product, and the distance function are elucidated. If the Finsleroid--Finsler space is of Landsberg type, then the Finsleroid charge is a constant. The Finsleroid--Finsler space proves to be a Berwald space if and only if the Finsleroid--axis 1-form is parallel with respect to the associated Riemannian metric and, simultaneously, the Finsleroid charge is a constant. The necessary and sufficient conditions for the Finsleroid--Finsler sp…

Finsleroid--Finsler Space and Spray Coefficients

In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular spray coefficients, we demonstrate how the Landsberg condition can explicitly appear in case of the Finsleroid--type metric function. Calculations are supplementing by a convenient special Maple--program. The general form of the associated geodesic spray coefficients is presented for such metric function under the condition of constancy of the Finsleroid charge…

On Concircularly Recurrent Finsler Manifolds

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 the present paper: 1. A concircularly flat Finsler manifold is necessarily of constant curvature (Theorem A); 2. Every $R^h$-recurrent Finsler manifold is concirculaly recurrent with the same recurrence form (Theorem B); 3. Every horizontally integrable concircularly recurrent Fi…

On Finsler surfaces of constant flag curvature with a Killing field

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 surfaces of constant flag curvature. In particular, we obtain the normal form of the Funk metric on the unit disk D^2.

Finsleroid-regular space developed. Berwald case

The Finsleroid--Finsler space becomes regular when the norm $||b||=c$ of the input 1-form $b$ is taken to be an arbitrary positive scalar $c(x) < 1$. By performing required direct evaluations, the respective spray coefficients have been obtained in a simple and transparent form. The adequate continuation into the regular pseudo-Finsleroid domain has been indicated. The Finsleroid-regular Berwald space is found under the assumptions that the Finsleroid charge is a constant and the 1-form $b$ is parallel. Keywords: Finsler metrics, spray coefficients, curvature tensors.

Hilbert forms for a Finsler metrizable projective class of sprays

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of spra…

Steiner's formula and a variational proof of the isoperimetric inequality

We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main difficulty in all of the proofs Steiner outlined for the plane isoperimetric inequality.

What is wrong with the Hausdorff measure in Finsler spaces

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily minimal, filling results such as those of S. Ivanov do not hold, and integral-geometric formulas do not exist. On the other hand, using the Holmes-Thompson definition of volume, we prove a general Crofton formula for Finsler spaces and give an easy proof that their totally geode…

Projectively Flat Finsler Metrics of Constant Curvature

It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on a bounded convex domain in R^n such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). Thus it is a natural problem to study those of constant curvature. In this paper, we study the Hilbert Fourth…

Reconstruction of Crofton Measures from Projective Finsler Metrics in R3

A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.

Concircular $π$-Vector Fields and Special Finsler Spaces

The aim of the present paper is to investigate intrinsically the notion of a concircular $π$-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Some properties of concircular $π$-vector fields are obtained. Different types of recurrence are discussed. The effect of the existence of a concircular $π$-vector field on some important special Finsler spaces is investigated. The whole work is formulated in a coordinate-free form.

Finsleroid-Finsler Space of Involutive Case

The Finsleroid-Finsler space is constructed over an underlying Riemannian space by the help of a scalar $g(x)$ and an input 1-form $b$ of unit length. Explicit form of the entailed tensors, as well as the respective spray coefficients, is evaluated. The involutive case means the framework in which the characteristic scalar $g(x)$ may vary in the direction assigned by $b$, such that $dg=μb$ with a scalar $μ(x)$. We show by required calculation that the involutive case realizes through the $A$-special relation the picture that instead of the Landsberg condition $\dot A_{ijk}=0$ we have the vanis…

Explicit formulae for geodesics in left invariant sub-Finsler problems on Heisenberg groups via convex trigonometry

In the present paper, we obtain explicit formulae for geodesics in some left-invariant sub-Finsler problems on Heisenberg groups $\mathbb{H}_{2n+1}$. Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2-dimensional sets, all left-invariant sub-Riemannian structures on $\mathbb{H}_{2n+1}$, and unit balls in $L_p$-metric for $1\le p\le\infty$. In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.

Volume forms for time orientable Finsler spacetimes

The paper proposes extensions of the usual notions of Finslerian volume to time orientable Finsler spacetime manifolds. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact, generally leading to infinite integrals) with integration on ellipsoids. Under certain conditions regarding the time orientations of the given spacetime, these ellipsoids can be attached to the given Lorentzian Finsler metric by means of a variational procedure. While the constr…

Invariant volume forms and first integrals for geodesically equivalent Finsler metrics

Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial.

几乎Finsler流形的特征张量

Characteristic tensors for almost Finsler manifolds

本文引入了几乎Finsler流形和部分Finsler流形，扩展了标准Finsler流形定义，允许非平凡切口包含在同源缩放下固定的点，并且允许基本张量具有非正特征值。通过双部空间给出了较为简单的几乎Finsler流形和部分Finsler流形的例子，并探讨了其物理应用。

On Asanov's Finsleroid-Finsler metrics as the solutions of a conformal rigidity problem

Finsleroid-Finsler metrics form an important class of singular (y-local) Finsler metrics. They were introduced by G. S. Asanov [2] in 2006. As the special case of the general construction Asanov produced singular (y - local) examples of Landsberg spaces of dimension at least three that are not of Berwald type. The existence of regular (y - global) Landsberg metrics that are not of Berwald type is an open problem up to this day; for a detailed exposition of the so-called unicorn problem in Finsler geometry see D. Bao [3]. In this paper we are going to characterize the Finsleroid-Finsler metrics…