Boundary rigidity proven for non-reversible Finsler metrics.
problem Recovering non-reversible Finsler metrics from boundary distance data.
method Sum of reversible Finsler norm and closed 1-form, boundary rigidity results.
result 1-form can be uniquely recovered from boundary distance data.
New metrics for Anosov representations defined from Thurston's asymmetric metrics.
problem Defining metrics for Anosov representations.
method Generalizing Thurston's asymmetric metric to Anosov representations.
result Provides a (possibly asymmetric) Finsler distance in some cases.
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.
Extends metric to Margulis spacetimes for convex properties.
problem No specific problem stated; extends metric.
method Extends Thurston's asymmetric metric to Margulis spacetimes and proves convex properties.
result Established convex properties of the extended metric.
Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …
The paper examines curvature-dimension bounds on sub-Finsler Heisenberg groups.
problem Investigating synthetic curvature-dimension bounds in sub-Finsler Heisenberg groups.
method Study of measure contraction property (MCP) and curvature-dimension condition (CD).
result Sub-Finsler Heisenberg groups do not satisfy MCP or CD for any parameters.
Characterizes isometries between non-reversible Finsler manifolds.
problem Understanding isometries in non-reversible Finsler manifolds.
method Generalization of Myers-Nakai Theorem for Riemannian manifolds, modification of function spaces to accommodate asymmetric structure.
result Functional characterization of isometries between non-reversible Finsler manifolds.
The paper studies horofunction compactifications of symmetric cones under Finsler distances.
problem Understanding horofunction compactifications of symmetric cones under Finsler distances.
method Establishing a correspondence between horofunction compactifications of symmetric cones and normed spaces, using Thompson and Hilbert distances.
result Explicit extensions of the exponential map and characterizations of horofunctions for Thompson and Hilbert distances.
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
Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge--Ampère type, whose correspondi…
The paper examines torsions in Minkowskian product of Finsler metrics.
problem Investigating Cartan torsion and mean Cartan torsion in Minkowskian product of Finsler metrics.
method Deriving explicit formulas for Cartan torsion and mean Cartan torsion, analyzing their geometric behavior, and providing conditions for bounded norm.
result Both Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean, and a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case is provided.
Paper proves rigidity theorems for geodesically reversible Finsler metrics.
problem Understanding geodesically reversible Finsler metrics in closed manifolds.
method Applied theory of volumes and areas on Finsler spaces to establish rigidity theorems.
result Partial explanation of the scarcity of geodesically reversible Finsler metrics in closed manifolds.
New Brownian motion defined in Minkowski normed spaces.
problem Constructing Brownian motion in non-Euclidean spaces.
method Singular McKean--Vlasov stochastic differential equation.
result Pathwise uniqueness of solutions to the stochastic differential equation.
Study shows horofunction compactification's topology matches dual norm's unit ball.
problem Global topology of horofunction compactification of Finsler manifolds.
method Construct explicit homeomorphisms for various spaces.
result Horofunction compactification homeomorphic to dual norm's unit ball.
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.
In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete Ck Finsler manifold M is determined by the normed algebra Cbk(M) of all real-valued, bounded and Ck smooth functions with bounded derivative defined on M. As a consequence, we obtain: (i) the Finsler structu…
The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.
problem The failure of the curvature-dimension condition in sub-Finsler geometry.
method Non-trivial adaptation of Juillet's work, introduction of new tools and ideas.
result The CD(K,N) condition does not hold in sub-Finsler geometry for various norms and measures. Convexity proven for sums of angles of unitary paths.
problem Proving convexity of sums of eigenvalues of unitary matrices.
method Analyzing paths of unitary matrices and their angles, using operator norms.
result Sum of first m angles of unitary path is convex.
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…
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
We study the structure of the stable norm of Finsler metrics on the 2-torus with a focus to points of irrational slope. By our results, the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Moreover, we study the stable norm in some natural examples.
We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifold…
Study circumcenters in Finsler unitary groups with optimal convexity bounds.
problem Existence and convexity of circumcenters in Finsler unitary groups.
method Analysis of distance functions and p-Schatten norm on Lie algebra.
result Existence of circumcenters for sets with radius < π/2 in several metrics.
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [5] and ha…
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.
problem Defining a norm and metric on Teichmüller spaces for surfaces of arbitrary genus.
method Adapting Thurston's earthquake norm to Riemann surfaces with marked points and using complex Legendre transforms.
result Establishes a complete analogue of Thurston's earthquake norm in the conformal setting.
Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.
problem Characterizing biholomorphic mappings between tube domains and bounded symmetric domains.
method Analyzing properties of Finsler symmetric cones and unital JB-algebras.
result Tube domains over Finsler symmetric cones are biholomorphic to bounded symmetric domains.
In this paper, the Cartan tensors of the (α,β)-norms are investigated in details. Then an equivalence theorem of (α,β)-norms is proved. As a consequence in Finsler geometry, general (α,β)-metrics on smooth manifolds of dimension n≥4 with vanishing Landsberg curvatures must be Berwald manifolds.
We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such me…
In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove…
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.
For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…
In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective n-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…
In this paper, we study locally projectively flat Finsler metrics with constant flag curvature K. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when K=0, K=−1 and K=1 are given respectively in an algebraic way.…
Examples of area-minimizing graphs with low regularity in a specific group.
problem Finding area-minimizing graphs with low regularity in a sub-Finsler Heisenberg group.
method Providing examples of entire area-minimizing horizontal graphs with prescribed singular sets.
result Examples of area-minimizing graphs that are locally Lipschitz but not necessarily smoother.
The paper studies connections and Finsler geometry on JB-algebra structure groups.
problem Investigating geometric structures on JB-algebra structure groups.
method Endowing the structure group with a connection and Finsler metric, computing quantities, and proving minimality of paths.
result Established the Finsler metric and distance on the cone of a JB-algebra.
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…
The paper characterizes compatible linear connections on 3D Finsler manifolds.
problem Characterizing compatible linear connections on Finsler manifolds of dimension three.
method Intrinsic method to characterize compatible linear connections, focusing on indicatrices and Euclidean symmetries.
result If a compatible linear connection is not unique, indicatrices must be Euclidean surfaces of revolution.
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.
Introduces new geodesic fields for Finsler manifolds.
problem Finding optimal paths in Finsler manifolds.
method Introduced Pontryagin type C0-Finsler structures and defined geodesic fields. result Extended geodesic field E provides more natural geodesics. The paper studies metrics on hyperkähler manifolds using sub-twistor constraints.
problem Understanding metrics on hyperkähler manifolds with non-holonomic constraints.
method Introducing and analyzing sub-conic metrics, showing they are sub-Finsler.
result Explicit description of Finsler norms for sub-twistor metrics.
Study smooths Finsler structures on Lie groups, proving extremal convergence.
problem Smooth left-invariant strongly convex C0-Finsler structures on Lie groups. method Introduce mollifier smoothing, study extremals using Pontryagin maximum principle.
result Pontryagin extremals on smoothed Finsler structures converge uniformly to those on original structure.
In this paper we study a sub-Finsler geometric problem on the free-nilpotent group of rank 2 and step 3. Such a group is also called Cartan group and has a natural structure of Carnot group, which we metrize considering the ℓ∞ norm on its first layer. We adopt the point of view of time-optimal control theory…
A C0-Finsler structure is a continuous function F:TM→[0,∞) defined on the tangent bundle of a differentiable manifold M such that its restriction to each tangent space is an asymmetric norm. We use the convolution of F with the standard mollifier in order to construct a mollifier smoothing of …
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. 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.