Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
Survey of recent metric geometry in Kähler metrics space.
problem Understanding the metric geometry of Kähler metrics space.
method Survey and highlighting of recent results.
result Highlighting of open problems in the field.
The paper finds maximal metrics on Euclidean spaces.
problem Finding maximal elements in moduli spaces of Riemannian metrics.
method Defining a preorder on moduli space by isometry groups and identifying maximal elements.
result Constructs many examples of maximal metrics on Euclidean spaces.
We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.
Quantum isometry groups extend to all countable metric spaces, and loose embeddings help understand metric space relationships.
problem Understanding the quantum isometry groups of all countable metric spaces.
method Defining and studying loose embeddability, showing that 0-dimensional compact metric spaces are generically loosely embeddable into the real line.
result 0-dimensional compact metric spaces are generically loosely embeddable into the real line.
Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.
problem Metric on space of Hermitian metrics on complex vector bundles.
method Compute metric spray, geodesics, curvature, and use Nash-Moser theorem.
result Metric completion of Hermitian metrics space is L2 integrable singular Hermitian metrics.
The paper characterizes Einstein metrics using CPE metrics and vacuum static spaces.
problem Understanding the conditions for Einstein metrics using CPE metrics and vacuum static spaces.
method Analyzing CPE metrics and their relationship with Einstein metrics and vacuum static spaces.
result A necessary and sufficient condition for a CPE metric to be Einstein in terms of σ_2-singular spaces is provided.
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. We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…
This thesis surveys various metrics on Riemann surface spaces.
problem Various metrics on Riemann surface spaces.
method Survey of metrics and their properties.
result Equivalence of Kähler-Einstein metric to Teichmüller metric.
New metric on geodesic currents connects different surface genera.
problem Understanding geodesic currents on surfaces of varying genera.
method Introducing a new asymmetric metric on the space of projective filling geodesic currents.
result Metric spaces of projective filling geodesic currents for surfaces of different genera are not isometric.
Defines curvature for metric triples in metric spaces.
problem No standard curvature for metric triples in general metric spaces.
method Defines curvature kX(T) using side lengths and distances to points in X. result Curvature kX(T) enables isometric embedding into model spaces. Completes the space of vector-valued one-forms on manifolds.
problem Metric incompleteness of the space of full-ranked one-forms.
method Distance equality and quotient structures.
result Concrete description of the metric completion of the space of full-ranked one-forms.
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
Generalizes Thurston's asymmetric metric to flat metrics.
problem Defining an asymmetric metric on flat metrics.
method Defined an asymmetric metric on the space of unit-area flat metrics.
result Discussed two different topologies from the asymmetry.
The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.
problem Analyzing conical Kähler-Einstein metrics on rank one horosymmetric Fano manifolds.
method Investigates the convergence of these metrics as the cone angle approaches zero, focusing on the behavior on the complement of a codimension one orbit.
result The metrics converge to the Kähler-Einstein metric on the basis and Stenzel's Ricci flat Kähler metrics on the fibers.
The paper describes geometric properties of Teichmüller space metrics.
problem Analyzing the geometry of Teichmüller space with weak Finsler metrics.
method Geometric description of unit spheres in weak Finsler metrics.
result Introduced a family of weak Finsler metrics interpolating between Thurston's metric and Teichmüller metric.
Study on moduli space of bi-invariant metrics in Lie groups.
problem Describing the space of bi-invariant metrics in Lie groups up to isometry.
method Showed BI is an orbifold and provided an explicit description. result Moduli space of bi-invariant metrics is an orbifold.
The notion of a Douglas space of second kind of a Finsler space with (α,β)-metric was introduced by I. Y. Lee [9]. Since then, so many geometers have studied this topic e. g., [14]. In this paper, we prove that a Douglas space of second kind with special -metric α+εβ+kαβ2 is conformally trans…
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. The paper introduces two new metrics on outer space and shows fixed points for their actions.
problem Analyzing metrics on outer space and their geometric group theory implications.
method Defined and analyzed entropy and pressure metrics on outer space, comparing to Weil-Petersson metric.
result For rank r≥4, the metrics have fixed points in their actions on outer space. Geodesic orbit metrics proven on specific homogeneous spaces.
problem Geodesic orbit metrics on homogeneous spaces.
method Using strongly isotropy irreducible spaces and proving natural reductivity.
result Geodesic orbit metrics are naturally reductive on constructed homogeneous spaces.
This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics …
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Finite approximations help reconstruct countable metric and ultrametric spaces.
problem Reconstructing countable metric and ultrametric spaces.
method Topological reconstruction using inverse limits of finite T0 spaces. result Countable metric and ultrametric spaces can be reconstructed as finite approximations.
Decomposes ultrametric spaces into scaled simplices.
problem Understanding the structure of ultrametric spaces.
method Introducing metric resolutions and coarse disjoint union.
result Constructs universal spaces for asymptotic dimension 0.
Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
Estimates means in metric spaces using quantization.
problem No practical estimator for Fréchet means in all metric spaces.
method Introduced estimators based on random quantization and data-driven partitioning.
result Universal consistency of estimators across separable metric spaces and Banach spaces.
Study of flat metrics on orbifolds and their moduli spaces.
problem Understanding flat metrics on orbifolds and their moduli spaces.
method Analysis of Teichmüller spaces and mapping class groups.
result Moduli space of flat metrics on orbifolds is a very good orbifold under certain conditions.
Paper characterizes embeddability of function spaces into Lp-type RKBS via metric entropy.
problem Characterizing embeddability of function spaces into Lp-type RKBS. method Establishes a connection between metric entropy growth and embeddability.
result A bound on metric entropy growth allows embedding into Lp-type RKBS. Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
Completed classification of Einstein spaces with specific metric properties.
problem Classifying Einstein spaces with a specific type of Stackel metric.
method Invariant under three-parameter abelian group of motions, completed classification of vacuum and electrovacuum spaces.
result Complete list of metrics for Einstein spaces in privileged coordinate systems.
We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that the weak homotopy type of this space is preserved by certain surgeries on the bou…
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Study shows how nonstandard hulls can contain metric completions.
problem Characterizing the Heine-Borel property in metric spaces.
method Using nonstandard analysis and metric completions.
result Characterization of the Heine-Borel property in terms of inapproachable finite points.
The paper finds two types of metric lines in curve spaces.
problem Classifying metric lines in jet spaces of curves.
method Established the existence of two families of metric lines in the 2-jet space of plane curves.
result Found precise criteria for identifying metric lines in sub-Riemannian geodesics.
Defines semi-symmetric metric connection on super warped products.
problem Computing curvature and Ricci tensors on super warped products.
method Introduced semi-symmetric metric connection and conditions for Einstein spaces.
result Conditions for super warped product spaces to be Einstein with semi-symmetric metric connection.
We study the Lipschitz metric on Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of Teichmuller space the Lipschitz metric is approximated up to bounded additive distortion by the sup metric on a product of lower-dimensional spaces (similar to the Teichm…
Study on spaces of metrics with intermediate curvature bounds.
problem Understanding spaces of metrics with lower bounds on intermediate curvatures.
method Analyzing spaces of Riemannian metrics with specific curvature bounds on high-dimensional Spin-manifolds.
result Spaces of metrics with positive p-curvature and k-positive Ricci curvature have non-trivial homotopy groups.
In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichmüller t…
Transforms metric space geometry into persistent homology.
problem Geometric properties of metric spaces encoded by curvature inequalities.
method Persistent homology induced by Čech filtration.
result Translation of geometric properties into topological representation.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
Using the identification of the symmetric space SL(n,R)/SO(n) with the Teichmüller space of flat n-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of t…
The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure …
Researchers extend the concept of metric spaces to Lorentzian spaces and prove the feasibility of their c-completion.
problem Extending the concept of metric spaces to Lorentzian spaces and proving their c-completion.
method Revisiting Lorentzian metric spaces, constructing c-completion, proving feasibility and endowing with Lorentzian metric space structure.
result The c-completion of Lorentzian metric spaces is feasible and well-suited, completing the original space in a precise sense.
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…
Researchers develop weighted GJMS operators for smooth metric measure spaces.
problem Developing mathematical tools for smooth metric measure spaces.
method Constructing and proving formal self-adjointness of weighted GJMS operators.
result Formal self-adjointness of weighted GJMS operators proved.
Study on stability of non-diagonal Einstein metrics on specific homogeneous spaces.
problem Stability analysis of non-diagonal Einstein metrics on HimesH/ΔK. method Formula for scalar curvature, study of stability with Hilbert action.
result Non-diagonal Einstein metrics on M are unstable with different coindexes.