Entropy rigidity for Finsler flows but collapse for Reeb flows.
problem Entropy behavior of Reeb and Finsler flows on contact manifolds.
method Analysis of topological entropy for Reeb and Finsler flows.
result Uniform positive lower bound for Finsler flows but arbitrarily small topological entropy for Reeb flows.
Introduces Finsler entropy for smooth distributions and foliations.
problem Entropy of smooth distributions and foliations.
method Definition of Finsler entropy based on distances on compact metric sets.
result Nullity of Finsler entropy for controllable distributions and singular Riemannian foliations.
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
New Finsler flow on 2-torus has chaotic dynamics.
problem Constructing chaotic dynamics on a 2-torus.
method Using Berger and Turaev's theorem, constructing a Finsler metric.
result Found a Finsler geodesic flow with positive metric entropy.
We show that a small perturbation of the boundary distance function of a simple Finsler metric on the n-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…
We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of foliated manifold with leafwise Randers structure. In this case the estimate…
Extends finite entropy measures in Kähler geometry.
problem Analyzing finite entropy measures on compact Kähler manifolds.
method Defining finite p-entropy and demonstrating their inclusion in an energy class. result Stability result for the complex Monge-Ampère equation.
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 4λ1>h2, where λ1 is the bottom of the L2-spectrum and h the…
We formulate a statistical analogy of regular Lagrange mechanics and Finsler geometry derived from Grisha Perelman's functionals generalized for nonholonomic Ricci flows. There are elaborated explicit constructions when nonholonomically constrained flows of Riemann metrics result in Finsler like configurations, and inv…
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…
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…
Researchers estimate gradients of solutions to a Finslerian Allen-Cahn equation.
problem Estimating gradients of solutions to a specific type of partial differential equation.
method Using the Finslerian Allen-Cahn equation as an Euler-Lagrange equation to a Liapunov entropy functional, proving gradient estimates on compact and noncompact Finsler metric measure spaces.
result Global and local gradient estimates of positive solutions to the Finslerian Allen-Cahn equation.
We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian ge…
We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weig…
We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci cur…
We present two approaches to the heat flow on a Finsler manifold (M,F): either as gradient flow on L2(M,m) for the energy; or as gradient flow on the reverse L2-Wasserstein space P2(M) of probability measures on M for the relative entropy. Both approaches depend on the choice of a measure m on …
We introduce a new distance metric for non-linear embeddings of Tempered Exponential Measures.
problem Non-linear embeddings of Tempered Exponential Measures (TEMs).
method Parameterization of finite discrete TEMs via Legendre functions, introducing tempered Hilbert co-simplex distance.
result Established a generalization of the Hilbert log cross-ratio simplex distance to a tempered Hilbert co-simplex distance.
We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …
One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the …
Study compares Calabi and Mabuchi geometries on Kähler manifolds and applies to flows.
problem Comparing metric geometries on Kähler manifolds and their applications.
method Introduced and explored Lp,q-Calabi Finsler structure; focused on finite entropy space. result The Lp-Calabi and L1-Mabuchi topologies coincide on the finite entropy space. Introduces singular Finsler foliations and their properties.
problem Generalizing Finsler actions, submersions, and foliations.
method Definition and analysis of singular Finsler foliations on Randers manifolds.
result Singular Finsler foliations on Randers manifolds are singular Riemannian foliations on the Riemannian manifold induced by the metric.
New types of special Finsler spaces studied.
problem Understanding new Finsler spaces.
method Pullback approach to global Finsler geometry.
result Properties and relations of new Finsler spaces.
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.
Study on Finsler spaces with vanishing torsion.
problem Energy-momentum conservation in Finsler gravity theories.
method Obtained a divergence theorem for pseudo-Finsler spaces with vanishing mean Cartan torsion.
result Clarified the problem of energy-momentum conservation in Finsler gravity theories.
The paper studies Finsler spaces with semi-concurrent vector fields and their equivalence to Riemannian spaces.
problem Characterizing Finsler spaces with semi-concurrent vector fields.
method Analyzing various Finsler spaces and proving conditions for equivalence to Riemannian spaces.
result Various Finsler spaces (quasi-C-reducible, C3-like, Ch-recurrent, P2-like) are equivalent to Riemannian spaces if they admit a semi-concurrent vector field. PhD dissertation on Finsler geometry and gravity, focusing on Berwald spaces and exact solutions.
problem Characterizing and solving Finsler gravity equations.
method Analysis of Berwald spaces, (α,β)-metrics, and exact solutions to Finsler gravity equations. result Exact vacuum solutions in Finsler gravity.
Smooth Busemann functions found in harmonic Finsler spaces.
problem Analyzing Busemann functions in Finsler manifolds.
method Investigation of Busemann functions in general and asymptotically harmonic Finsler manifolds.
result Smoothness of Busemann functions on asymptotically harmonic Finsler manifolds.
Researchers generalize cosmological models using Finsler geometry.
problem Cosmological models that closely approximate pseudo-Riemannian geometry.
method Identifying Lie Algebra of symmetry generators for spatially homogeneous and isotropic Finsler geometries.
result Found the most general spatially homogeneous and isotropic Berwald spacetimes.
The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.
problem Estimating distance functions and proving Schwarz lemma for weakly Kähler-Finsler manifolds.
method Establishing theorems about distance functions and applying them to prove the Schwarz lemma.
result Holomorphic mappings from weakly Kähler-Finsler manifolds to pseudoconvex Finsler manifolds are constant under certain conditions.
The paper proves a theorem about Finsler manifolds and their diameters.
problem Understanding the maximum diameter of Finsler manifolds.
method Analyzing weighted Ricci curvature and applying it to the diameter.
result Finsler manifolds with maximal diameter are isometric to standard Finsler spheres.
Study of Hamilton's Ricci flow on Finsler spaces, proving existence and convergence.
problem Existence and convergence of Hamilton's Ricci flow on Finsler spaces.
method Defined and proved existence of Finslerian Ricci-DeTurck flow, used to find Hamilton's Ricci flow solution.
result Existence of short time solution to Hamilton's Ricci flow on Finsler spaces.
The study introduces new tensors for almost Finsler manifolds and analyzes their properties.
problem Defining and analyzing new types of Finsler manifolds.
method Introducing new Finsler manifolds, studying their properties, and deriving characteristic tensors.
result Characteristic tensors for almost Finsler manifolds have been generalized and their properties have been studied.
New method for calculating Finsler connections using moving frame formalism.
problem Calculating Finsler connections for metrics depending on moving frame.
method Developed a moving frame formalism to simplify Finsler connection calculations.
result Derived formulae for Finsler connections when metric depends on moving frame.
The present paper deals with the Killing correspondence between some Finsler spaces. We consider a Finsler space equipped with a β-change of metric and study the Killing correspondence between the original Finsler space and the Finsler space equipped with β-change of metric. We obtain necessary and sufficient condi…
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. 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.
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 proves unique Finsler connections for scalar forms.
problem Existence and uniqueness of Finsler connections.
method Pullback approach to global Finsler geometry, study of horizontally recurrent connections.
result Existence and uniqueness of horizontally recurrent Finsler connections for scalar forms.
Study on geodesics in 2D generalized Finsler spaces, focusing on singularities.
problem Analyzing geodesic singularities in two-dimensional generalized Finsler spaces.
method Defined geodesics as extremals of an auxiliary functional, considering isotropic lines as geodesics.
result Identified geodesic singularities in two-dimensional generalized Finsler spaces.
The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
A theorem on Finsler metrics with special curvature properties.
problem Characterizing Finsler metrics with sectional flag curvature.
method Analyzing the relationship between flag curvature and sectional curvature.
result Locally, Finsler metrics are either Riemannian or have isotropic flag curvature.
Paper classifies positively curved δ-homogeneous Finsler metrics.
problem Classifying positively curved Finsler metrics.
method Exploring δ-homogeneity and using approximation techniques.
result Classification of positively curved δ-homogeneous Finsler spaces.
The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and su…
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.
The paper discusses fundamental problems in global Finsler geometry.
problem Fundamental problems in global Finsler geometry.
method Optimization and improvement of Lie derivatives, characterization of gradient vector fields, gradient estimate.
result Gradient estimate for smooth functions on Randers manifolds.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
The paper studies special Finsler spaces with Hp-scalar curvature.
problem Characterizing and investigating Finsler spaces with specific scalar curvatures.
method Intrinsic investigation and various conditions for transformations between Finsler spaces.
result Conditions for transforming Finsler spaces of scalar curvature to those of Hp-scalar curvature. The paper defines semi-concurrent vector fields on Finsler manifolds and explores their implications.
problem Characterizing Finsler manifolds with semi-concurrent vector fields.
method Introducing and investigating semi-concurrent vector fields, proving properties, and providing examples.
result Finsler manifolds admitting semi-concurrent vector fields are either Riemannian or conic Finslerian.