Thomas Barthelmé
In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4λ_1 > h^2$, where $λ_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow.
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Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian metric on $M$ and $β$ an appropriate $1$-form on $M$, it is shown that the first eigenvalue $λ_{1,p}(M,F)$ of the Finslerian $p$-Laplacian defined by the Finsler metric $F$ is controled by the first eigenvalue $λ_{1,p}(M,g)$ of the Riemannian $p$-Laplacian defined on $(M,g)$. Fi…
By using Bochner technique and gradient estimate, we give the lower bound estimates of the first eigenvalue of Finsler-Laplacian on Finsler manifolds. These results generalize the corresponding famous theorems in the Riemannian geometry.
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 obtain some partial results in this direction when the Finsler metric is Berwald.
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。
Operators on nonlinear metric measure spaces I: A New Laplacian comparison theorem on Finsler manifolds and a generalized Li-Yau approach to gradient estimates of Finslerian Schrödinger equation
本文在 Finsler 流形上引入混合加权 Ricci 曲率并证明新的 Laplacian 比较定理，据此建立 Finsler Schrödinger 方程正解的整体与局部 Li-Yau 型梯度估计。
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 the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue.
In this paper, we study some potential theoretic aspects of the eikonal and infinity Laplace operator on a Finsler manifold $M$. Our main result shows that the forward completeness of $M$ can be detected in terms of Liouville properties and maximum principles at infinity for subsolutions of suitable inequalities, including $Δ^N_\infty u \ge g(u)$. Also, an $\infty$-capacity criterion and a viscosity version of Ekeland principle are proved to be equivalent to the forward completeness of $M$. Part of the proof hinges on a new boundary-to-interior Lipschitz estimate for solutions of $Δ^N_\infty u…
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 Finsler manifolds.
On Liouville problems for $(p,q)$-Laplacian inequalities on Finsler measure spaces
该研究针对芬斯勒测度空间上的两类$(p,q)$-拉普拉斯不等式，在势函数满足积分增长条件时证明非负弱解几乎处处为零，核心方法为非线性容量法。
We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.
Finsler Geometry, Graph Neural Networks, and You
针对图拉普拉斯只能近似各向同性算子的局限，提出基于芬斯勒拉普拉斯的图神经网络层，证明其收敛性并恢复非线性扩散方程的几何结构。
We prove the Bochner-Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.
The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of critical Kähler Finsler metrics is obtained. A Yamabe type problem for mean Ricci curvature is considered.
On Finsler metric measure manifolds with integral weighted Ricci curvature bounds
本文研究具有积分加权Ricci曲率界的Finsler度量测度流形，建立多个比较定理，得到体积增长估计、Gromov预紧性及Dirichlet等周常数估计，并应用于特征值与调和函数梯度估计。
Isoperimetric inequality on Finsler metric measure manifolds with non-negative weighted Ricci curvature
本文在具有非负加权 Ricci 曲率的 Finsler 度量测度流形上，通过定义体积熵和第二 Cheeger 常数，证明了精确等周不等式，并以此推导出 Finsler Laplacian 第一特征值的 Cheeger-Buser 型不等式。
In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler-Laplacian operator, we introduce \textit{faithful dimension pairs} by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik-Schnirelmann category, Krasnoselskii genus and essential …
We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective …
We show that monochromatic Finsler metrics, i.e., Finsler metrics such that each two tangent spaces are isomorphic as normed spaces, are generalized Berwald metrics, i.e., there exists an affine connection, possibly with torsion, that preserves the Finsler function
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduced $hh$-curvature on finite time. Next, it is shown that the evolution of Ricci scalar is a parabolic-type equation and if the initial Finsler metric is of positive flag curvature, then the flag curvature and the Ricci scalar remain positive as long as the solution exists. Finall…
In our previous work, we have defined a nonlinear connection of Finsler manifold which preserves the Finsler metric $L=L(x,dx)$. To make the method easier and more useful in applications, moving frame (vielbein) $θ^a={e^a}_μdx^μ$ formalism for the nonlinear connection is newly considered. We derive formulae to calculate the Finsler connection in the specific case that the Finsler metric depends not on coordinates $x^μ$, but only on moving frame $θ^a$: $L=L(θ)$.
The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry
本文系统发展了任意符号伪芬斯勒度量的Palatini形式体系，通过引入独立非线性连接构造的里奇标量导出了仿射与度量方程，并在特定条件下恢复了经典结论。
New classes of Finsler metrics: The birth of new projective invatiant
本研究在芬斯勒几何中提出新的射影不变量，通过构建W-GDW方程将GDW度量推广为更广泛的W-GDW度量类，含两个子类，阐明了这类度量的性质并辅以示例。
The general notion of anisotropic connections $\nabla$ is revisited, including its precise relations with the standard setting of pseudo-Finsler metrics, i.e., the canonic nonlinear connection and the (linear) Finslerian connections. In particular, the vertically trivial Finsler connections are identified canonically with anisotropic connections. So, these connections provide a simple intrinsic interpretation of a part of any Finsler connection closer to the Koszul formulation in $M$. Moreover, a new covariant derivative and parallel transport along curves is introduced, taking first a self-pr…
Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds. Compatible linear connections are the solutions of the so-called compatibility equations containing the torsion compone…
Analysis on asymmetric metric measure spaces: $q$-heat flow, $q$-Laplacian and Sobolev spaces
该研究将对称度量测度空间的几何分析概念推广至非对称情形，引入上梯度、$q$-拉普拉斯算子等，研究$q$-热流与Sobolev空间，并用不可逆Finsler流形构建示例揭示两类框架的本质差异。
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in term…
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald …
Connections and Finsler geometry of the structure group of a JB-algebra
本文研究无限维JB-代数结构群的左不变联络与Finsler度量，通过将正锥表示为齐性空间，证明了单参数群及某些路径的极小性。
Anisotropic conformal change of conic pseudo-Finsler surfaces, I
本文研究圆锥伪Finsler曲面的各向异性共形变换，利用修改的Berwald标架推导出逆度量张量分量的张量形式变换，并探讨了变换后保持伪Finsler性质的条件及几何不变量。
A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the unicity of the linear connection in general, the first step of checking the existence of compatible linear connections on a Finsler manifold is to choose the best one to look for. A reasonable choice is introduced in \cite{V14} called the extremal compatible linear connection, w…
On Generalized Matsumoto Metrics with a Special $π$-form
本文研究了广义Matsumoto度量的构造及其几何性质，探讨了其与原度量的关系及不变性条件。
The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Borel, non negative and locally finite. ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $Δg=μ$, where $g$ i…
Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations
本文通过局部共形形变构造能量势垒，证明 Finsler 流形上存在任意多条连接两点的测地线，并推广到 Randers 度量、电磁轨迹及光线，得到多重性结果。
A Concurrent Generalized Kropina Change
本文研究具特定π-形式的广义Kropina度量，从容许并发π-向量场的芬斯勒流形出发考察φ-并发广义Kropina变换，分析其几何对象、给出实例，证明喷雾非射影相关等性质及变换保持几乎有理性质。
Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with o…
Given a compact Riemannian manifold (M, g) and two positive functions $ρ$ and $σ$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $σ$, with respect to the L 2 inner product weighted by $ρ$. Under some regularity conditions on $ρ$ and $σ$, these eigenvalues are those of the operator $ρ$^{-1} div($σ$$\nabla$u) with Neumann conditions on the boundary if $\partial$M = $\emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.
Hadamard type variation formulas for the eigenvalues of the $η$-Laplacian and applications
本文研究了η-拉普拉斯算子特征值的Hadamard型变分公式，证明了在特定条件下特征值通常简单，并探讨了其应用。
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