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.
Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.
problem Establishing gradient estimates for the Finslerian Fisher-KPP equation.
method Global gradient estimates on compact and noncompact Finsler metric measure spaces using the traditional CD(K,N) condition and new comparison theorems. result Global gradient estimates for positive solutions of the Finslerian Fisher-KPP equation.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
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.
The paper is devoted to weighted Lp-Hardy inequalities with best constants on Finsler metric measure manifolds. There are two major ingredients. The first, which is the main part of this paper, is the Hardy inequalities concerned with distance functions in the Finsler setting. In this case, we find that besides the …
The paper proves uncertainty principles on Finsler measure spaces.
problem Uncertainty principles on Finsler measure spaces.
method Analyzes Lp-uncertainty principles on Finsler measure spaces with bounded curvatures. result Sharp Lp-uncertainty principles are proven and characterized. Researchers prove constant solutions for a specific Finslerian equation.
problem Investigating exponentially harmonic functions on Finslerian spaces.
method Analyzing the exponential energy functional and using nonnegative Ricci curvature conditions.
result Any bounded solution to the Finslerian equation is constant.
Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
Paper develops methods for estimating gradients of Finslerian Schrödinger equations.
problem Estimating gradients of Finslerian Schrödinger equations.
method Develops new Laplacian comparison theorem and applies it to Finslerian Schrödinger equation.
result Global and local Li-Yau type gradient estimates for positive solutions.
We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the L1-gradient estimate on Finsler manifolds (under some additional assumptions in the n…
Study bounds on curvature for special Finsler metrics.
problem Curvature and topological properties of ∞-Einstein Finsler metrics. method Construct special metrics, analyze equivalence, impose curvature bounds.
result Establish bounds for curvature and distortion on ∞-Einstein Finsler manifolds. Sharp log-Sobolev inequalities proved for CD(0,N) spaces.
problem Proving log-Sobolev inequalities in noncompact metric measure spaces.
method Sharp isoperimetric inequality, symmetrisation, scaling argument, Hamilton-Jacobi inequality, Sobolev regularity.
result Sharp log-Sobolev inequalities established in CD(0,N) spaces. The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
Researchers describe horofunctions in noncompact Hermitian symmetric spaces.
problem Understanding horofunctions in noncompact Hermitian symmetric spaces.
method Realized noncompact Hermitian symmetric spaces as open unit balls in Banach spaces with Jordan structures.
result Complete description of horofunctions in the metric compactification.
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.
This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface ar…
Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.
problem Infinitesimal Hilbertianity of sub-Riemannian manifolds with general measures.
method Embedding metric derivations into square-integrable sections, approximating sub-Finsler distances.
result Sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary Radon measures.
The paper solves the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.
problem Solving the isoperimetric problem on noncompact metric spaces with lower Ricci bounds.
method Establishes a structure theorem for minimizing sequences, proving the limit of such sequences is identified by a finite collection of isoperimetric regions.
result The limit of a minimizing sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space.
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
The paper derives gradient estimates for solutions of certain equations on metric measure spaces.
problem Gradient estimates for solutions of specific nonlinear and elliptic equations on metric measure spaces.
method Derives Li-Yau and Hamilton's type gradient estimates for positive solutions.
result Gradient estimates for positive solutions of the equations on complete noncompact metric measure spaces.
Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
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…
In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler-Einstein metrics while real metric measure spaces are considered with Bakry-Émery Ricci tensor. There are tw…
We highlight several analogies between the Finsler (infinitesimal) properties of Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space. Thurston defined his asymmetric metric in analogy with Teichmüllers' metric, as a solution to an extremal problem, which consists, in the case of the asymmetric me…
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.
Noncompact RCD spaces with maximal first Betti number are rigid.
problem Characterizing noncompact RCD spaces with maximal first Betti number.
method Analyzing properties of noncompact RCD spaces with maximal first Betti number.
result Spaces with maximal first Betti number are either flat Riemannian manifolds or metric products.
The paper studies solutions to a nonlinear equation on Finsler manifolds with gradient estimates and Harnack inequalities.
problem Exploring positive solutions to a nonlinear parabolic equation on Finsler manifolds.
method Developed new comparison theorems and used Li-Yau estimates.
result Established gradient estimates and Harnack inequalities for solutions.
We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant Finsler metric on X. As an application, we establish the existence of natural bordifications, as orbifold…
Study Ricci curvature of homogeneous Finsler spaces with specific metrics.
problem Curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Derived explicit formulae for Ricci curvature and found conditions for vanishing S-curvature. result Spaces with vanishing S-curvature and negative Ricci curvature are Riemannian. Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Study flag curvature in homogeneous Finsler spaces with a specific metric.
problem Analyzing flag curvature in homogeneous Finsler spaces with a generalized m-Kropina metric. method Provided explicit formula for flag curvature, showed equivalence of definitions, and studied curvature of naturally reductive spaces.
result Equivalence of two definitions of naturally reductive homogeneous Finsler spaces for the generalized m-Kropina metric. We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov sp…
Proves rectifiability for specific metric spaces with unique tangents.
problem Rectifiability of CD(K,N) and MCP(K,N) spaces with unique tangents. method Failure of CD condition in sub-Finsler Carnot groups, new result on MCP spaces, recent breakthrough by Bate. result Proves rectifiability for CD(K,N) and MCP(K,N) spaces under specific conditions. 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.
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…
In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold …
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…
A new metric framework for weighted projective spaces improves clustering and analysis.
problem Proximity measurement in weighted projective spaces with intrinsic scaling and topology.
method Hierarchical clustering framework based on Finsler geometry, quotienting weighted scaling action.
result The constructed metric dF satisfies the triangle inequality, making it a genuine metric. Busemann G-spaces with Finsler metrics
problem Admitting a differentiable DC atlas with a continuous Finsler metric
method Comparison geometry
result Generalizes previous work on G-spaces with Riemannian curvature bounds
The study defines Finsler metrics on special surfaces and constructs geodesic currents.
problem Defining and studying Finsler metrics on specific geometric structures.
method Defined compatible Finsler distances, studied geodesics, and constructed Liouville currents.
result Constructs a Liouville current for each metric, encoding curve lengths.
New definition of metric current yields Finsler geometry volume densities.
problem Defining volume functionals from Finsler geometry.
method Proposed a new definition of metric current and showed its utility.
result Obtained a family of extendibly convex volume densities.
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.
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 formu…
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…
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…
The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.
problem Characterizing almost Ricci solitons on Finsler measure spaces.
method Introducing and investigating gradient almost Ricci solitons, proving conditions for existence.
result Conditions for the existence of gradient almost Ricci solitons on Finsler measure spaces.