Study on curvature equation in Heisenberg group with convex boundary.
problem Existence of solutions to prescribed mean curvature equation in sub-Finsler Heisenberg group.
method Finsler approximation scheme to prove existence of Lipschitz solutions.
result Existence of a Lipschitz solution for the Dirichlet problem.
Let us consider a Riemannian manifold M (either separable or non-separable). We prove that, for every ε>0, every Lipschitz function f:M→R can be uniformly approximated by a Lipschitz, C1-smooth function g with $\Lip(g)\le \Lip(f)+ε$. As a consequence, every Riemannian manifold is uniformly …
In this paper, we explore the similarity between normal homogeneity and δ-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected δ-homogeneous Finsler space is G-δ-homo-geneous, for some suitably chosen connected quasi-compact G. So δ-homogeneous Fins…
Revisits Finsler spacetimes from inertial observer perspective.
problem Physical foundations of relativistic spacetimes.
method Inertial observers and double linear approximation.
result Finsler spacetimes are defined by dropping the second linearization.
The tangent space is constructed in sub-Finsler geometry, leading to the failure of the CD condition in 3D-contact manifolds.
problem The failure of the CD condition in sub-Finsler geometry.
method Construction of the tangent space in the measured Gromov-Hausdorff sense, application of nilpotent approximation.
result The CD condition fails in 3D-contact sub-Finsler manifolds.
Study of strictly accretive matrices using Finsler geometry.
problem Characterize the set of strictly accretive matrices.
method Introduced Finsler metrics and characterized geodesics and distance.
result Geodesic distance applied to matrix approximation problem.
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all we…
Stability result for nearly isometric subspaces and Finsler surfaces.
problem Stability of normed spaces and Finsler surfaces under near-isometric conditions.
method Refined topological argument and explicit quantification using Banach-Mazur distance.
result A 2-dimensional surface with near-monochromatic Finsler metric is approximately Riemannian.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
Modeling wildfire spread using Gielis superformula and Finsler spacetime.
problem Accurately modeling the short-time spread of wildfires.
method Using Gielis superformula and Finsler spacetime to determine firefronts.
result A concise and efficient expression of geodesic equations.
The paper extends Sobolev spaces to Finsler manifolds and shows density of smooth functions.
problem Defining and analyzing Sobolev spaces on Finsler manifolds.
method Defining Sobolev spaces for Finsler structures and proving density of smooth functions.
result Smooth functions can approximate weak solutions and functions in Sobolev spaces on Finsler manifolds.
The space of Kähler metrics can, on the one hand, be approximated by subspaces of algebraic metrics, while, on the other hand, can be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former s…
New Finsler metrics describe trace function growth rates in convex projective surfaces.
problem Understanding growth rates of trace functions in convex projective surfaces.
method Introduced new Finsler metrics and showed their convergence to describe trace function growth.
result Logarithms of trace functions are approximated by lengths in a Finsler metric defined by cubic differential.
This paper studies gradient flows in asymmetric metric spaces and proves existence results.
problem Investigating gradient flows in asymmetric metric spaces.
method Discrete approximation and natural convexity assumption on potential function.
result Existence of curves of maximal slope in asymmetric metric spaces.
We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of H1-curves, and two splitting lemmas for the functionals on Banach manifolds of C1-curves. Two results on critical groups of iterated closed geodesics a…
New method approximates anisotropic curve shortening flow.
problem Approximating anisotropic curve shortening flow.
method Weak formulation and finite element approximation.
result Optimal H1-error bound for approximation. Let (M,F) be a C∞ Finsler manifold, p≥1 a real number, k a positive integer and Hkp(M) a certain Sobolev space determined by a Finsler structure F. Here, it is shown that the set of all real C∞ functions with compact support on M is dense in the Sobolev space H1p(M). This resul…
GEORCE computes geodesics quickly and accurately.
problem Computing geodesics on Riemannian and Finsler manifolds is difficult and inefficient.
method GEORCE transforms geodesic computation into a discrete control problem.
result GEORCE achieves global convergence and quadratic local convergence.
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.
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.
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 F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then $\mathcal{F}…
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 study solves the isoperimetric problem for Heisenberg group norms.
problem Solving the isoperimetric problem for anisotropic norms in the Heisenberg group.
method Representation formula for perimeter, foliation property, differential equation characterization, approximation procedure.
result Characterization of isoperimetric sets as sub-Finsler analogues of Pansu's bubbles.
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.
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…
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.
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.
The pullback approach to global Finsler geometry is adopted. Some new types of special Finsler spaces are introduced and investigated, namely, Ricci, generalized Ricci, projectively recurrent and m-projectively recurrent Finsler spaces. The properties of these special Finsler spaces are studied and the relations betwee…
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…
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.
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 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.
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.
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 Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.
problem Constructing Funk-Finsler structures in hyperbolic models.
method Using Finsler isometries and explicit computations, the Funk-Finsler structure is constructed in various hyperbolic models.
result The Funk-Finsler structure in the Klein unit disc is a Randers metric.
The study extends Obata's theorem and classifies Finsler manifolds with transnormal functions.
problem Classifying Finsler manifolds based on geometric properties.
method Extending Obata's theorem and using a second order differential equation.
result Complete Finsler manifolds of positive constant flag curvature are homeomorphic to spheres.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
The paper shows examples of geodesics switching infinitely often on certain manifolds.
problem Understanding geodesics with infinitely many switches on Finsler and sub-Finsler manifolds.
method Provided examples and explicit structures on Carnot groups, presented a sufficient condition for chattering.
result Geodesics on certain manifolds can exhibit a countable number of switches in arbitrarily small time intervals.
In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…
Study on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
The paper proves properties of Finsler submanifolds and analytic maps.
problem Analyzing properties of Finsler submanifolds and their analytic maps.
method Proving properties of regular fibers of analytic maps and Finsler submersions.
result Regular fibers of an analytic map are equifocal under certain conditions.
New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first eigenvalue of the Finsler-Laplacian attains its lower bound if and only if the Fi…
The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.
problem Analyzing curvature properties of spherical Finsler metrics.
method Proving semi-C-reducibility and finding conditions for vanishing mean stretch curvature.
result Conditions for a general spherically symmetric Finsler metric to have vanishing mean stretch curvature.