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-quasi hyperbolic metric in convex geometry. Uniform estimates for Kaehler metrics' diameters and volumes.
problem Estimating diameters and volumes of Kaehler metrics.
method Proving uniform diameter and volume estimates for a family of Kaehler metrics.
result Uniform estimates for diameters and volumes of Kaehler metrics.
Estimates for scalar curvature equations on Kähler manifolds with singularities.
problem Developing estimates for scalar curvature equations with singular metrics.
method Estimates and Laplacian estimates for scalar curvature equations of degenerate Kähler metrics.
result Derivation of estimates for singular constant scalar curvature Kähler metrics and singular Kähler-Einstein metrics.
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
Estimates Kaehler metrics' diameter in big cohomology classes.
problem Estimating the diameter of Kaehler metrics in big cohomology classes.
method Proves uniform diameter estimates using integrability conditions and stability properties of complex Monge-Ampere equations.
result Uniform diameter estimates for Kaehler metrics in big cohomology classes.
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.
Estimates scalar curvature of point clouds without embedding.
problem Estimating scalar curvature of data sets without embedding.
method Intrinsic estimator based on metric structure, consistent and stable.
result Estimator converges to scalar curvature as sample size increases.
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
Sharp ABP estimate on metric spaces via optimal transport.
problem Sharp ABP estimate on metric measure spaces.
method Optimal transport theory.
result Established a sharp ABP estimate on metric measure spaces.
Study geodesic Lie groups' convergence to limits with quantitative estimates.
problem Quantifying convergence rates of geodesic Lie groups to their limits.
method Estimates on the difference between original metrics and asymptotic/tangent metrics.
result Sharpens existing bounds on convergence rates.
Estimates gradients of solutions on closed surfaces.
problem Gradient estimates for solutions on closed surfaces.
method Considered a new metric g′=e2ug with bounded integral curvature, derived gradient estimates for g′, and used these to obtain gradient estimates for u. result Gradient estimates for solutions on closed surfaces are established.
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.
We study the minimax optimal rate for estimating the Wasserstein-1 metric between two unknown probability measures based on n i.i.d. empirical samples from them. We show that estimating the Wasserstein metric itself between probability measures, is not significantly easier than estimating the probability measures u…
Estimates for harmonic forms on a 3-Torus, proving their existence.
problem Existence of nowhere vanishing harmonic 1-forms on a 3-Torus.
method Explicit computation of injectivity estimates using the Laplace operator on the 3-Torus and its perturbations.
result Existence of a nowhere vanishing harmonic 1-form on a perturbed metric on the 3-Torus.
The accurate measurement of security metrics is a critical research problem because an improper or inaccurate measurement process can ruin the usefulness of the metrics, no matter how well they are defined. This is a highly challenging problem particularly when the ground truth is unknown or noisy. In contrast to the w…
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.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
Uniform Sobolev inequality for Kähler metrics with entropy bound.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.
We refine a metric bunching estimate for pinched manifolds.
problem Improving an unstable bunching estimate for pinched metrics.
method Compact Riemannian manifolds with pointwise negatively pinched curvature tensor.
result Improved unstable bunching estimate.
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 proves estimates for Hermitian metrics and shows curvature blow-up on complex manifolds.
problem Estimating curvature blow-up in Hermitian metrics.
method Local Calabi and higher order estimates for continuity equations.
result Chern scalar curvature blows up at a finite-time singularity on compact complex manifolds.
Proves estimates for Calabi-Yau metrics as Kahler classes shrink.
problem Estimates for Calabi-Yau metrics under shrinking Kahler classes.
method Proves asymptotic expansion in terms of powers of fiber diameter with uniform C^k-estimates.
result Uniform estimates for all orders of derivatives of Calabi-Yau metrics.
Paper refines Chen-Cheng's estimates for Kähler metrics.
problem Uniform boundedness of scalar curvature assumption.
method Replacing uniform boundedness with Lp-boundedness. result Improved estimates for Kähler metrics under Lp-boundedness. The study finds counterexamples to curvature estimates for minimizing surfaces.
problem Curvature estimates for minimizing surfaces in metric convergence.
method Constructing sequences of smooth minimizing surfaces in metrics converging to Euclidean.
result Found counterexamples with diverging L2 norm of second fundamental form. This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kähler metrics ωε when ε→0. Here ωε are flat and have areas ε and ε−1 on the two elliptic curves …
We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which …
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.
The paper provides gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.
method Using Hamilton type and Li-Yau type estimates, the paper proves gradient estimates on positive solutions to generalized nonlinear parabolic equations on smooth metric measure spaces with compact boundary.
result Gradient estimates for nonlinear heat-type equations on smooth metric measure spaces.
Estimates mass of static vacuum metrics with small Bartnik data.
problem Estimating mass of static vacuum metrics with small perturbations.
method Second-order mass estimation using Bartnik data.
result New upper bound on Bartnik mass to fifth order.
Improved Kuznecov remainder estimates for generic metrics.
problem Estimating period integrals of Laplace eigenfunctions on manifolds.
method Two-term asymptotic expansion and elimination of oscillatory second term.
result Improved remainder estimates for Baire-generic metrics.
A new perceptual adjustment query for metric learning reduces complexity in high-dimensional data.
problem Metric learning in high-dimensional data with limited human feedback.
method Inverted measurement scheme and two-stage estimator for PAQs.
result Sample complexity guarantees for the two-stage estimator of metric learning from PAQs.
Estimates Bartnik mass for metrics with nonnegative Gauss curvature.
problem Estimating Bartnik mass for specific metric configurations.
method Using area, total mean curvature, and a metric roundness measure.
result Estimate approaches sharp value for round spheres.
New estimates for Green's functions in varying Kähler metrics.
problem Uniform estimates for Green's functions in Kähler metrics.
method Broadening techniques to allow complex structure variation and removing assumptions.
result Uniform estimates for Green's functions in families of canonical Kähler metrics.
This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.
problem Boundedness conditions for Kähler metrics with bounded entropy and scalar curvature.
method Slightly relaxes the boundedness condition on the scalar curvature.
result Apriori estimates and C3,α estimate for the potential of the Kähler metrics under relaxed conditions. The regularity theory for pluriclosed flow hinges on obtaining Cα regularity for the metric assuming uniform equivalence to a background metric. This estimate was established in \cite{StreetsPCFBI} by an adaptation of ideas from Evans-Krylov, the key input being a sharp differential inequality satisfied by the assoc…
The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.
problem Gradient estimates for solutions to a specific nonlinear elliptic equation on smooth metric measure spaces.
method Nash-Moser iteration technique to obtain local gradient estimates.
result New local gradient estimates for positive solutions to the equation.
New metric reduces estimation error in survival model evaluation.
problem Dependent censoring complicates survival model evaluation.
method Dependent Brier score based on Archimedean copula and Copula-Graphic estimator.
result Reduces estimation error by 12-16% on average.
New approaches estimate recommendation metrics using sampling.
problem Understanding and resolving the use of sampling for recommendation evaluation.
method MLE and ME principles for empirical rank distribution recovery.
result Advantages of new approaches for top-k metrics estimation.
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
Estimates scalar curvature on moduli space as genus grows.
problem Estimating scalar curvature on moduli space.
method Analyzing Weil-Petersson metric for large genus.
result Precise estimate of average scalar curvature up to 1/g2. Estimates metric tensor on neuromanifolds using Fisher information and random methods.
problem Computing the metric tensor on high-dimensional neuromanifolds efficiently and accurately.
method Deterministic bounds and unbiased random estimators based on Hutchinson's trace method.
result An efficient random estimator with bounded standard deviation.
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
New metrics improve uncertainty estimation on graph data.
problem Current GNNs focus only on nodewise scores, limiting uncertainty estimation.
method Proposed edgewise metrics for uncertainty estimation on graphs.
result GNN models with structured prediction perform better in uncertainty estimation.
New eigenvalue bounds for 3-Sasaki metrics improve previous estimates.
problem Estimating eigenvalues for 3-Sasaki metrics.
method Improved Lichnerowicz-Obata type estimates for scalar sub-Laplacian.
result Lower bounds for the first non-zero eigenvalue of 3-Sasaki metrics.
Research uses CPS to estimate uncertainty in ML radio metric models.
problem Estimating uncertainty in machine learning models for radio metrics and path loss.
method Conformal Prediction (CP) in Conformal Predictive Systems (CPS) with diverse difficulty estimators.
result CPS models maintain high coverage and reliability across different cities.
Accurately estimating uncertainties in neural network predictions is of great importance in building trusted DNNs-based models, and there is an increasing interest in providing accurate uncertainty estimation on many tasks, such as security cameras and autonomous driving vehicles. In this paper, we focus on the two mai…
The Liouville theorem and Cα-estimate for Calabi-Yau cones establish uniqueness and asymptotic behavior of metrics.
problem Establishing uniqueness and asymptotic behavior of metrics on Calabi-Yau cones.
method Developed a Liouville theorem and C0,α-estimate for Ricci-flat, conical Kähler manifolds. result Uniformly bounded Kähler metrics on a ball around the apex are asymptotic to the Ricci-flat cone metric with polynomial decay.