Uniform K-stability ensures existence of special metrics on toric manifolds.
problem Existence of conformally Kähler, Einstein-Maxwell metrics on toric manifolds.
method Introducing uniform K-stability and showing its equivalence to properness of relative K-energy.
result Uniform K-stability is necessary and sufficient for the existence of f-extremal metrics on toric manifolds. Surveying recent work on Kähler metrics and algebraic variety stability.
problem Understanding canonical Kähler metrics on algebraic varieties.
method Analyzing recent developments in algebraic geometry.
result Relation between canonical Kähler metrics and stability in algebraic geometry.
We can talk about two kinds of stability of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional F. The other one is a dynamical stability and it refers to a convergence of a Ricci flow starting at any metric in a neighbourhood of a considere…
Paper examines stability of minimizing metrics on manifolds with boundary.
problem Stability of minimizing Yamabe metrics on compact manifolds with boundary.
method Investigates stability in the sense introduced by Escobar, showing closeness of nearly minimizing metrics.
result Quantitative closeness of nearly minimizing metrics to minimizing Yamabe metrics within their conformal class.
Study stability and instability of Ricci-flat metrics under generalized Ricci flow.
problem Stability and instability of Ricci-flat metrics under Ricci flow.
method Analysis of generalized Ricci flow for Ricci-flat metrics and vanishing 3-forms.
result Dynamical stability and instability results for Ricci-flat metrics and vanishing 3-forms.
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
New stability theorem for hyperbolic metrics without volume bounds.
problem Stability of finite volume hyperbolic metrics without upper volume bounds.
method Abstract axiomatic framework and bootstrap argument to extend stability result.
result Weaker exponential control of the metric allows for a broader application of the stability theorem.
Surveying stability and deformation of Einstein metrics.
problem Stability and deformation of Einstein metrics.
method Study of the spectrum and eigentensors of the Lichnerowicz Laplacian.
result Recent results on stability and deformation theory of Einstein metrics.
Flow stabilizes on non-Kähler metrics near Calabi-Yau.
problem Stability of conformally balanced metrics flow near Calabi-Yau manifolds.
method Proving stability of the anomaly flow around Calabi-Yau metrics.
result The flow can converge on non-Kähler metrics near Calabi-Yau.
The paper proves stability for Einstein metrics with special twisted spinors.
problem Stability of Einstein metrics with specific spinor conditions.
method Proves linear semi-stability for a class of Einstein metrics with non-positive scalar curvature.
result Linear semi-stability for Einstein metrics carrying a parallel twisted spinr spinor. Introduces stability conditions for polarized varieties, linking to K-stability.
problem Stability conditions for polarized varieties.
method Analogue of Bridgeland's stability for polarized varieties, Z-stability, Z-critical Kähler metrics.
result Polarized varieties with certain stability conditions admit Z-critical Kähler metrics.
We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compac…
Paper addresses travel time tomography stability and statistical inversion.
problem Determining conformal factors of metrics from geodesic lengths.
method Established forward and inverse stability estimates; applied to Bayesian statistical inversion.
result Consistency of statistical inversion technique for travel time tomography.
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.
problem Existence of constant scalar curvature Kähler metrics on polarized manifolds.
method Direct proof using microscopic stability thresholds and conditions on the limit.
result Existence of a unique constant scalar curvature Kähler metric under specific conditions.
We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain …
New Einstein metrics identified on a specific full flag manifold.
problem Investigating Einstein metrics on a full flag manifold.
method Analyzing G-stability of Einstein metrics on M=G/K. result Identified four new Einstein metrics on F(5), confirming pairwise non-homotheticity. Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Dona…
From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We…
Study on special Hermitian metrics and their stability.
problem Existence and stability of Hermitian metrics with specific properties.
method Analysis of Hermitian metrics with $∂ar{∂}ω^k=0$ for k=1 to n−1. result Stability of metrics at blow-up and deformations.
Defines new stability conditions for Sasaki manifolds and extremal metrics.
problem Stability conditions for Sasaki manifolds and extremal metrics.
method Combining weighted K-stability with Sasaki extremality theory.
result Weighted K-stability is necessary for extremal Sasaki metrics.
We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of E…
Article proves effective conditions for existence of Kähler metrics.
problem Existence of extremal Kähler metrics on fibrations.
method Weighted uniform K-stability conditions derived from moment polytopes.
result Various effective conditions for K-stability verified.
Hermitian-Einstein metrics linked to stability of bundles on orbifolds.
problem Existence of Hermitian-Einstein metrics on stable vector bundles over compact Kähler orbifolds.
method Equivalence of slope stability to the existence of Hermitian-Einstein metrics and properness of a functional.
result Equivalence of Hermitian-Einstein metrics and slope stability for stable vector bundles.
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Log K-polystability and G-uniform log K-stability are established. result Uniform log K-stability is achieved for normal varieties. We prove that the existence of extremal metrics implies asymptotically relative Chow stability. An application of this is the uniqueness, up to automorphisms, of extremal metrics in any polarization.
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. We study the linear stability of Einstein metrics of Riemannian submersion type. First, we derive a general instability condition for such Einstein metrics and provide some applications. Then we study instability arising from Riemannian product structures on the base. As an application, we estimate the coindex of the E…
Quantitative stability for nearly minimizing Yamabe metrics.
problem Understanding the stability of nearly minimizing metrics in Riemannian geometry.
method Proving quantitative closeness of nearly minimizing metrics to minimizing metrics in a specific sense.
result The distance between nearly minimizing metrics and minimizing metrics is controlled quadratically by the Yamabe energy deficit.
Simplified proof of stability for Ricci flow near ALE metrics.
problem Stability of Ricci flow near ALE metrics with integrable deformations.
method Equivalence between integrability and almost-orthogonality property of Ricci-DeTurck tensor, analysis in weighted Holder spaces.
result Dynamical stability of Ricci flow near linearly stable Ricci-flat ALE metrics.
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under a…
Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.
problem Understanding K-stability and Kähler-Einstein metrics for cubic fourfolds.
method Local volume estimates and Ambro-Kawamata's non-vanishing theorem.
result All smooth cubic fourfolds admit Kähler-Einstein metrics.
New SPD metrics improve stability and efficiency in neural networks.
problem Designing stable and efficient Riemannian metrics on SPD manifolds.
method Cholesky decomposition to derive SPD metrics.
result Proposed metrics provide closed-form operators, computational efficiency, and improved numerical stability.
New approach to Z-stability and critical metrics on Kähler manifolds.
problem Determining Z-stability and existence of Z-critical metrics on Kähler manifolds. method Equivariant localisation applied to integrals over test configurations.
result Existence of Z-critical metrics is equivalent to Z-stability. Paper defines new stability and metrics for complex spaces.
problem Stability and metrics for complex spaces with big cohomology classes.
method Introduces slope stability and Hermitian-Einstein metrics for big cohomology classes.
result Kobayashi Hitchin correspondence and Bogomolov Gieseker inequality proved.
Develops harmonic metrics for Hull-Strominger system stability.
problem Existence of solutions to the Hull-Strominger system with balanced class.
method Uses non-Hermitian Yang-Mills connections and holomorphic Courant algebroids, introduces harmonic metrics.
result Expected existence of a numerical stability condition for generic families of solutions.
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
Proves existence of Kähler-Einstein metrics with certain twisting forms.
problem Existence of Kähler-Einstein metrics under stability conditions.
method Proves existence assuming twisted K-stability condition, allows certain non-negative twisting forms.
result Existence of twisted Kähler-Einstein metrics under specified conditions.
Stability of submanifold cut loci under metric perturbations proved.
problem Stability of submanifold cut loci under metric perturbations.
method Continuity of injectivity radius and Whitney C2 perturbation of submanifolds. result Hausdorff stability of submanifold cut loci under C2 metric perturbations. The paper proves stability for scalar-flat metrics on manifolds with boundary.
problem Stability of scalar-flat metrics on manifolds with boundary.
method Reduced problem to boundary functional and used deficit control.
result Deficit controls distance to minimizing set on manifolds with boundary.
Stability of cut locus under metric perturbations in compact Riemannian manifolds.
problem Stability of cut locus under C2-perturbations of the metric. method Proving stability with respect to the Hausdorff metric of the cut locus under C2 perturbation of the metric. result The Hausdorff distance between cut loci converges to zero as the metrics converge.
The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.
problem Linear stability of Gradient Ricci Shrinker metrics.
method Found commutator formulas and generalized a stability theorem.
result Generalized a necessary condition for linear stability.
The paper studies Yamabe metrics and stability in Riemannian manifolds.
problem Existence of complete Yamabe metrics with zero scalar curvature.
method Yamabe flow and local L1-stability analysis. result Local L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature. Stability result for a popular algorithm in optimal transport.
problem Stability of the Iterative Proportional Fitting Procedure in time and metric.
method Uniform stability analysis in the 1-Wasserstein metric.
result Quantitative stability result for entropy-regularized Optimal Transport and Schrödinger bridges.
In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.