The Simanca metric on a blown-up plane has regular quantization properties.
problem Characterizing the quantization properties of the Simanca metric.
method Using the blow-up structure and Tian-Yau-Zelditch expansion, proving regular quantization and vanishing coefficients.
result All coefficients in the Tian-Yau-Zelditch expansion for the Simanca metric vanish, and a dense subset admits Berezin quantization.
Classifies radial scalar flat metrics with a constant third coefficient.
problem Classifying radial scalar flat metrics with a specific coefficient.
method Analyzing the TYZ expansion and Simanca's scalar flat metric.
result Characterization of Simanca's scalar flat metric as a byproduct.
The paper studies scalar flat Kähler metrics on line bundles and proves their properties.
problem Understanding scalar flat Kähler metrics on line bundles.
method Analyzes two families of scalar flat Kähler metrics on C n + 1 \mathbb{C}^{n+1} C n + 1 and O ( − k ) \mathcal{O}(-k) O ( − k ) , proving existence of asymptotic expansions and approximations. result Characterizes the Burns-Simanca metric as the only projectively induced scalar flat metric on O ( − k ) \mathcal{O}(-k) O ( − k ) with a vanishing second coefficient in its asymptotic expansion. Study Kähler-Einstein manifolds with holomorphic isometries into blow-ups of complex spaces.
problem Characterizing Kähler-Einstein manifolds with holomorphic isometries into specific blow-ups.
method Analyzing properties of Kähler-Einstein manifolds and their isometries into generalized Burns-Simanca and Eguchi-Hanson manifolds.
result Generalized Burns-Simanca and Eguchi-Hanson manifolds are not relatives to any homogeneous bounded domain.
Two conjectures on Ricci-flat metrics verified under specific conditions.
problem Properties of Ricci-flat metrics on complex manifolds.
method Analyzing conjectures and verifying them under specific conditions.
result Conjectures verified for certain types of metrics on complex surfaces.
We investigate geometric invariants of the one parameter family of Mukai threefolds that admit C ∗ \mathbb C^{*} C ∗ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in t…
Proves uniform K-stability is open in Kähler cone.
problem Stability of Kähler metrics in complex geometry.
method Introduced new norm on test configurations and estimates for non-archimedean energy functionals.
result Uniform K-stability is an open condition in the Kähler cone.
Study on volumes and scalar curvature in complex geometry, proving new bounds and conditions.
problem Understanding volumes and scalar curvature in complex geometry.
method Asymptotic behavior of Bergman kernel, Kähler-Ricci flow, singular Kähler-Einstein metrics, and gluing techniques.
result New bounds and conditions for volumes and scalar curvature in complex manifolds.
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.
Study on conditions for Randers metrics to be of constant Ricci curvature.
problem Conditions for Randers metrics to be of constant Ricci curvature.
method Analysis of sufficient and necessary conditions for Randers metrics with and without strong convexity.
result Classification of Randers metrics with ∥ β ∥ α > 1 \|β\|_α>1 ∥ β ∥ α > 1 and ∥ β ∥ α ≡ 1 \|β\|_α\equiv1 ∥ β ∥ α ≡ 1 . 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.
New Finsler metrics constructed from G D W GDW G D W -metrics.
problem Exploring new Finsler metrics within the G D W GDW G D W -metric class. method Constructing new sub-classes of G D W GDW G D W -metrics. result Presented illustrative examples of new Finsler metrics.
The paper introduces new Finsler metrics and associated weighted quasi-metrics.
problem Geometric properties of Finsler metrics and quasi-metric spaces.
method Construction of weighted quasi-metrics associated with Finsler metrics.
result Investigation of geometric properties of weighted quasi-metric spaces.
We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
Introduces Finslerian convolution metrics and their properties.
problem No specific problem stated; focuses on new metric concept.
method Definition and study of Finslerian convolution metrics.
result Characterization of Finslerian convolution metrics of Riemannian, Minkowskian, and Randers types.
New balanced metrics introduced for SPD matrices, improving metric choice.
problem Lack of principles for choosing SPD matrix metrics.
method Introducing balanced metrics that relate existing metrics.
result Two new balanced metric families introduced: mixed-power-Euclidean and mixed-power-affine.
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 defines metrics from Lie groups and conjectures they are Einstein metrics.
problem Defining metrics from Lie groups.
method Analyzing metrics from specific Lie groups like unitary, orthogonal, and symplectic groups.
result Conjectures that metrics from these Lie groups are Einstein metrics.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
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…
New Kähler metrics generalize Calabi's and relate to Fano manifolds.
problem Existence of extremal Kähler metrics on Fano manifolds.
method Introducing σ σ σ -extremal Kähler metrics and relating them to multiplier Hermitian-Einstein metrics. result Existence of σ σ σ -extremal Kähler metrics implies existence of multiplier Hermitian-Einstein metrics on Fano manifolds. New Finsler metrics defined by Riemannian and 1-forms are studied.
problem Characterize and study properties of ( α , β , γ ) (α,β,γ) ( α , β , γ ) -metrics. method Introduced and defined ( α , β , γ ) (α,β,γ) ( α , β , γ ) -metrics, analyzed their properties, and found conditions for local projective flatness and Douglas type. result Necessary and sufficient conditions for ( α , β , γ ) (α,β,γ) ( α , β , γ ) -metrics to be locally projectively flat and Douglas type were found. We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kaehler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivati…
The paper examines ( α , β ) (α,β) ( α , β ) -metrics and proves they are Berwald metrics under certain conditions.
problem Characterizing ( α , β ) (α,β) ( α , β ) -metrics of weak Landsberg type. method Analyzing properties of ( α , β ) (α,β) ( α , β ) -metrics, proving conditions for weak Landsberg metrics to be Berwald. result Every weak Landsberg ( α , β ) (α,β) ( α , β ) -metric is a Berwald metric when β β β is closed and conformal. 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 S S -curvature and mean Landsberg curvature leading to vanishing curvature. Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k k k -quasi hyperbolic metric in convex geometry. The study examines Lee metrics on groups and their properties.
problem Characterizing groups that admit Lee metrics.
method Analyzing conditions for groups to have or not have Lee metrics, studying specific families of groups, and providing tables for groups of order ≤ 31.
result Conditions for groups to have Lee metrics, including specific families and non-cyclic groups.
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
Study shows convergence of Lagrangian submanifolds under certain metrics.
problem Understanding convergence of Lagrangian submanifolds under specific metrics.
method Proves convergence to an embedded Lagrangian submanifold using a monotonicity lemma applied on a carefully-chosen metric ball.
result Convergence to an embedded Lagrangian submanifold implies convergence in the Hausdorff metric for a class of metrics.
In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.
Defines a new Randers metric based on an existing one.
problem No specific problem stated; focuses on defining a new metric.
method Defines a new left-invariant Randers metric i l d e F ilde{F} i l d e F based on an existing one F F F . result Shows that F F F is of Berwald (Douglas) type if and only if i l d e F ilde{F} i l d e F is of Berwald (Douglas) type. The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.
problem Limited coverage of O(n)-invariant metrics by kernel metrics.
method Characterization of O(n)-invariant metrics, intermediate classes construction.
result Introduction of cometric-stability as a key property for geodesics.
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.
New ambient metrics reveal properties of Walker metrics.
problem Characterizing Walker metrics using ambient metrics.
method Developed Fefferman-Graham ambient metrics for Walker metrics.
result Walker metrics have vanishing Q-curvature.
New interpretation of metrics on special geometric manifolds.
problem Finding metrics on extremal Kähler manifolds.
method Moment map interpretation of relatively balanced metrics.
result Extremal metrics are limits of specific relatively balanced metrics.
Paper characterizes square metrics with vanishing Douglas curvature.
problem Characterizing square metrics with vanishing Douglas curvature.
method Using β-deformations to achieve analytical examples.
result Characterization of singular square metrics with vanishing Douglas curvature.
Proposes a method to select fair performance metrics through metric elicitation.
problem Choosing fair performance metrics in multiclass classification with multiple sensitive groups.
method Metric elicitation strategy that requires only relative preference feedback and is robust to noise.
result Elicits group-fair performance metrics for multiclass classification problems.
Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general ( α , β ) (α,β) ( α , β ) -metrics, which are defined by a Riemannian metric α = a i j ( x ) y i y j α=\sqrt{a_{ij}(x)y^iy^j} α = a ij ( x ) y i y j and a 1 1 1 -fo…
In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R 3 \boldsymbol{R}^3 R 3 are Kossowski metrics, and t…
In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of Finsler metrics of Einstein-reversibility are always Einstein metrics. In parti…
metric-learn simplifies metric learning in Python.
problem Performing distance metric learning efficiently.
method Unified scikit-learn compatible interface for supervised and weakly-supervised metric learning.
result Unified interface for cross-validation and model selection.
Introduces new metric for Riemannian metrics, extending unbalanced optimal transport.
problem Extending unbalanced optimal transport to Riemannian metrics.
method Dynamic and static formulations of unbalanced optimal transport on Riemannian metrics.
result Wasserstein--Ebin metric provides a new Riemannian structure on the space of Riemannian metrics.
The paper explores learning metrics in low dimensions with bounds and complexities.
problem Learning metrics in low dimensions with bounds and complexities.
method Develops upper and lower bounds on generalization error, quantifies sample complexity, and bounds accuracy relative to the true metric.
result Novel mathematical approaches to metric learning and insights into ordinal embedding.
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
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.