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.
The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…
Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.
problem Characterizing weakly weighted Einstein-Finsler metrics.
method Showed isotropic S-curvature under certain conditions. Characterized via navigation expressions and α and β. result Weakly weighted Einstein-Kropina metrics have isotropic S-curvature and can be completely characterized.
The paper studies weighted Ricci curvatures and characterizes Randers metrics.
problem Characterizing Randers metrics with weighted Ricci curvatures.
method General weighted Ricci curvatures and characterization of Randers metrics.
result Characterization of Randers metrics with almost isotropic weighted Ricci curvatures.
Proves existence of weighted-cscK metrics on Kähler manifolds.
problem Existence of weighted-cscK metrics on Kähler manifolds.
method Proves existence via G-coercivity of weighted Mabuchi functional. result Existence of (v, w)-weighted-cscK metrics with v log-concave.
Paper proves existence of weighted constant scalar curvature metrics.
problem Existence of weighted constant scalar curvature Kähler metrics.
method Coercivity of weighted Mabuchi functional implies existence of wcscK metric.
result Equivalence of coercivity and existence of wcscK metrics.
Uniqueness of weighted extremal metrics on Kähler manifolds proven.
problem Uniqueness of weighted extremal Kähler metrics on compact Kähler manifolds.
method Proof of uniqueness using modified Mabuchi energy and weighted K-semistability.
result Uniqueness of weighted extremal Kähler metrics up to automorphisms.
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
Study on weighted cscK metrics on Kähler varieties with singularities.
problem Existence of singular weighted cscK metrics on Kähler varieties.
method Resolution of singularities, coercive weighted Mabuchi functional, construction of examples.
result Existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive.
Blowups of Kähler manifolds can inherit extremal metrics.
problem Extending extremal metrics to blowups of Kähler manifolds.
method Analyzing the action of a torus on blowups and weighted extremal metrics.
result Blowups of Kähler manifolds can inherit weighted extremal metrics.
Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.
problem Extending classical extremal Kähler metrics to include new objects.
method Surveying recent works on weighted extremal Kähler metrics and the YTD conjecture.
result Survey of recent research on weighted extremal Kähler metrics.
In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted proj…
New mass and staticity concepts derived from weighted curvature maps.
problem Deriving mass and staticity concepts for weighted manifolds.
method Developed a weighted curvature map and its adjoint, leading to weighted mass and static metrics.
result Equivalence and uniqueness theorems for weighted static manifolds and Penrose inequality.
This paper connects soliton-type metrics with weighted CSCK metrics on Fano manifolds.
problem Existence and properties of weighted constant scalar curvature Kähler metrics.
method Introducing a weight function g(v,w) and proving equivalence between (v,w)-CSCK metrics and g(v,w)-solitons. result Existence of (v,w)-CSCK metrics in the first Chern class is equivalent to existence of g(v,w)-solitons. The paper studies invariant weighted Bergman metrics on domains.
problem Investigating invariant weighted Bergman metrics under biholomorphisms.
method Introducing invariant weight assignments, using Bergman's minimum integral method and domain version of Tian-Yau-Zelditch expansion.
result Uniform convergence of weighted Bergman kernels and metrics on uniform squeezing domains.
Classifies smooth metric measure spaces with two weighted Einstein representatives.
problem Classifying smooth metric measure spaces with specific weighted Einstein properties.
method Local and global classification using Einstein and quasi-Einstein warped products.
result Global classification result for complete manifolds, showing specific types of manifolds.
The paper predicts edge weights in weighted directed networks using metric geometry.
problem Predicting edge weights in weighted directed networks.
method Introducing new types of weighted directed networks (AWDNs), constructing metrics, and proposing modified kNN and SVM methods.
result The proposed methods outperform traditional approaches in predicting edge weights.
Establishes Yau-Tian-Donaldson conjecture for weighted metrics.
problem Constant scalar curvature Kähler metrics on polarized projective manifolds.
method Extends Chi Li's work to weighted case, uses a priori estimates and slope formulas.
result Proves Yau-Tian-Donaldson conjecture for weighted extremal Kähler metrics.
Survey on Kähler-Einstein and weighted solitons on Fano manifolds.
problem Existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds.
method Generalization of algebraic conditions for K-polystability.
result Existence of coupled Kähler-Einstein metrics and weighted solitons is equivalent to algebraic conditions.
Real-world machine learning applications often have complex test metrics, and may have training and test data that are not identically distributed. Motivated by known connections between complex test metrics and cost-weighted learning, we propose addressing these issues by using a weighted loss function with a standard…
We propose a definition of the weighted σk-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted σk-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when k=1,2 or the smooth metric measure space is lo…
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.
Novel metric space magnitude and weighting vectors improve machine learning tasks.
problem Improving machine learning algorithms using novel metric space concepts.
method Metric space magnitude and weighting vectors for better machine learning.
result The weighting vector effectively detects boundaries and improves classic machine learning tasks.
Paper develops a framework to optimize neural networks using weighted metrics.
problem Discrepancy between maximizing weighted classification scores and minimizing loss function.
method Formalizes weighted classification metrics and constructs corresponding losses.
result Framework includes well-established approaches like cost-sensitive learning and weighted cross entropy.
Extends online learning to metric spaces using exponential weights.
problem Online learning in metric spaces.
method Exponentially weighted average forecaster, barycenters, Jensen's inequality, measure contraction property.
result Results in a statistical learning framework.
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …
Study Bochner formula on metric measure spaces for vanishing Betti numbers.
problem Vanishing Betti numbers on metric measure spaces.
method Introduce weighted curvature conditions.
result Vanishing of all Betti numbers.
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 classifies Einstein spaces and warped products in weighted geometry.
problem Characterizing geometric structures of weighted Einstein spaces.
method Complete local classification of weighted Einstein spaces with harmonic Weyl tensor.
result Spaces decompose into Einstein or specific warped products.
Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.
problem Characterizing extremal Kahler and Sasaki metrics using energy coercivity.
method Maximal complex torus, coercive weighted Mabuchi energy, K-polystability.
result Coercive weighted Mabuchi energy implies strict positivity of Donaldson-Futaki invariant and existence of extremal metrics.
Adapts example weights to optimize black-box metrics.
problem Optimizing metrics defined by black-box functions.
method Adaptive example weighting and iterative post-shifting.
result Improves classification performance compared to baselines.
Study on Kähler metrics with curvature constraints.
problem Existence of constant weighted scalar curvature Kähler metrics.
method Establish Ck-estimates for Kähler potentials. result Extends prior results on classical cscK metrics.
We study the existence of weighted extremal Kähler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauducho…
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
We introduce a notion of a Kähler metric with constant weighted scalar curvature on a compact Kähler manifold X, depending on a fixed real torus T in the reduced group of automorphisms of X, and two smooth (weight) functions v>0 and w, defined on the momentum image (with respect to …
Proves invariance of weighted extremal Kähler metrics under smooth blowups.
problem Invariance of weighted extremal Kähler metrics under smooth blowups.
method Uniform coercivity estimate for the (relative, weighted) Mabuchi energy on blowups.
result Invariance of weighted extremal Kähler metrics under smooth blowups.
The weighted Yamabe flow converges on smooth metric measure spaces.
problem Analyzing convergence of the weighted Yamabe flow on metric measure spaces.
method Introduced the weighted Yamabe flow and proved its long-time existence and convergence under certain conditions.
result Long-time existence and convergence of the weighted Yamabe flow on smooth metric measure spaces.
Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric with cone singularities along the lines asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expr…
We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz …
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
We prove existence and uniqueness of weighted ambient metric for manifolds with density.
problem Existence and uniqueness of weighted ambient metric for manifolds with density.
method Proving existence and uniqueness of weighted ambient metric for manifolds with density.
result Existence and uniqueness of weighted ambient metric for manifolds with density.
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not a…
In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi metric, the K-energy, the degenerate complex Hessian equation. The second part is on…
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. The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
problem Conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
method Investigation of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
result Identification and determination of weight functions with real holomorphic gradient fields on specific metrics.