The paper classifies stable toric Fano manifolds and compares K-stability and Ding stability.
problem Classifying stable toric Fano manifolds in low dimensions.
method Using Mabuchi constants calculated from moment polytopes and Bott tower structure.
result List of uniform relative Ding stability for toric Fano manifolds up to four dimensions.
The study examines necessary conditions for Mabuchi solitons on Fano manifolds and their relation to Ding stability.
problem Existence of Mabuchi solitons on Fano manifolds.
method Investigates the inner product of C∗-actions on equivariant test-configurations and uses convex-geometry descriptions. result Uniformly relative Ding stability implies a necessary condition for the existence of Mabuchi solitons.
Study Mabuchi solitons on toric Fano varieties, linking stability and energy.
problem Existence and stability of Mabuchi solitons on toric Fano varieties.
method Algebraic stability notion (relative Ding stability) and variational approach.
result Partial coercivity and singular Mabuchi solitons in non-uniformly stable cases.
Characterizes stable toric Fano manifolds using modified Ding functional.
problem Stability of toric Fano manifolds.
method Characterization through modified Ding functional and pseudo-boundedness analysis.
result Characterization of relative Ding stable toric Fano manifolds.
The paper proves a new stability condition for certain Fano varieties.
problem Stability conditions for Fano varieties with Gorenstein singularities.
method Using toric test configurations and combinatorial criteria.
result Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties.
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.
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.
New stability criteria for Fano varieties using generalized b-divisors.
problem Characterizing uniform K-stability in Fano varieties. method Introducing a new function ildeδ and formalism for K-stability, proving stability conditions for Kähler-Einstein metrics. result Existence of a unique Kähler-Einstein metric implies uniform D-log K-stability when ildeδ(D)>1. New stability criterion for Fano manifolds using anticanonically balanced metrics.
problem Stability conditions for Fano manifolds and their invariant δm. method Proof of equivalence between stability condition and anticanonically balanced metrics.
result Established a Hilbert-Mumford type criterion for δm>1. In this paper, we study the limiting properties of the K energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In …
New approach finds analytic interpretation of algebraic invariants for balanced metrics.
problem Finding analytic interpretation of algebraic invariants for balanced metrics.
method Using log canonical thresholds and basis divisors, the approach involves quantized Ding functionals on Bergman spaces.
result Each δ_m is the coercivity threshold of a quantized Ding functional on the m-th Bergman space, characterizing the existence of balanced metrics.
We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existe…
Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.
problem Characterizing and approximating non-Archimedean metrics on pseudoeffective classes.
method Extending Ross-Witt Nyström correspondence to relative case, introducing flag configurations.
result Non-Archimedean finite energy metrics are approximable by flag configurations, and very general Ding energies are continuous.
We show that the coercivity of the modified Ding functional leads to the existence of a certain kind of balanced metrics and their convergence to the Kähler-Ricci soliton modulo automorphisms. In our results, we do not assume that the vanishing of the higher order modified Futaki invariants introduced by Berman-Nyström…
Researchers compute Hessian of Ding functionals and analyze its convexity and asymptotic behavior.
problem Analyzing the convexity and asymptotic behavior of quantized Ding functionals.
method Computed the Hessian of quantized Ding functionals using projective geometry and Berezin-Toeplitz quantization.
result Elementary proof of convexity of quantized Ding functionals along Bergman geodesics.
Study Mabuchi metrics on Fano manifolds proving their existence and properness.
problem Existence and properness of Mabuchi metrics on Fano manifolds.
method Prove existence using properness of modified Ding functional and inverse implication.
result Establish criterion for Mabuchi metrics existence on Fano group compactifications.
Suppose (X,J,ω) is a Fano manifold and t→rt is a diverging Kähler-Ricci trajectory. We construct a bounded geodesic ray t→ut weakly asymptotic to t→rt, along which Ding's F-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fi…
The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.
problem Existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. method Analyzes Q-Fano group compactifications, proving uniqueness and existence of Kähler-Einstein metrics. result Proves the existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. A mathematical model of a Ricci flow solution.
problem Modeling the three-dimensional standard solution to Ricci flow.
method Developed a Bryant soliton model.
result Validated the Bryant soliton model.
Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
problem Proving Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.
method Analyzing Monge-Ampère equations corresponding to generalized and twisted Kähler-Ricci g-solitons, proving stability conditions.
result Existence of solutions is equivalent to equivariantly uniform Θ-twisted g-Ding-stability.
Proves existence of Kähler-Einstein metrics in big cohomology classes.
problem Existence of Kähler-Einstein metrics in big cohomology classes.
method Using a divisorial stability condition and Fujita-Odaka type delta invariants, building up from scratch the theory of pluripotential theory.
result Uniform Yau-Tian-Donaldson existence theorem for Kähler-Einstein metrics in the big cohomology class setting.
New approach uses isomorphic vector fields for stability of relative equilibria.
problem Stability of relative equilibria in nonlinear systems.
method Introduced and used isomorphic vector fields to study stability.
result Obtained a criterion for nonlinear stability of relative equilibria.
Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.
problem Existence of Kähler-Ricci soliton degeneration for log Fano pairs.
method Algebraic proof of two-step degeneration to uniformly Ding stable triple.
result Log Fano pairs admit Kähler-Ricci soliton when ground field is complex.
Introduces relative stability conditions on triangulated categories.
problem Stability conditions in triangulated categories.
method Definition and deformation of relative stability conditions.
result Deformation of relative stability conditions via gluing stability conditions.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
The inverse Monge-Ampere flow helps find Kahler-Einstein metrics.
problem Finding Kahler-Einstein metrics on complex manifolds.
method Gradient flow of the Ding energy functional on Kahler metrics.
result The flow converges to a Kahler-Einstein metric in various cases.
In this paper we study the relative Chow and K-stability of toric manifolds in the toric sense. First, we give a criterion for relative K-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford cr…
Note: Ding-Jost-Li-Wang's result holds for non-constant h.
problem Proving J achieves minimum for non-constant h.
method Excluding blow-up points on h's zero set.
result Ding-Jost-Li-Wang's result extends to non-constant h.
Characterizes geometric actions on graphs with flexible stabilizers.
problem Understanding geometric actions on flexible stabilizers.
method Defining generalized fine actions and proving relative quasi-convexity criteria.
result Characterizes Bowditch boundary points in relatively geometric actions.
New proof given for a functional's minimum condition.
problem Functional's minimum condition under varying h. method Variational method and maximum principle.
result Functional achieves minimum under Ding-Jost-Li-Wang condition.
For a polarized algebraic manifold (X,L), let T be an algebraic torus in the group of all holomorphic automorphisms of X. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking T to be trivial, we see that asymptotic Chow-stability follows from stron…
Extremal metrics linked to stability in complex geometry.
problem Existence and uniqueness of extremal metrics in complex geometry.
method Proving asymptotic relative Chow stability implies extremal metrics existence and uniqueness.
result Existence and uniqueness of extremal metrics in any polarization.
The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
The study proves Kähler manifolds with extremal Kähler metrics are relatively K-stable.
problem Existence of extremal Kähler metrics on Kähler manifolds.
method Introducing relative K-stability and proving it for Kähler manifolds with extremal Kähler metrics.
result Kähler manifolds admitting extremal Kähler metrics are relatively K-stable.
Paper introduces Ricci iteration for studying Kähler-Einstein metrics.
problem Study of coupled Kähler-Einstein metrics.
method Coupled Ricci iteration, proving smooth convergence and coercivity.
result Smooth convergence of iteration for both negative and positive first Chern classes.
Mabuchi's metric correlates with a specific stability condition for Fano manifolds.
problem Characterizing Fano manifolds with Mabuchi's soliton metric.
method Proving Mabuchi's metric corresponds to relative D-stability.
result Fano manifolds admit Mabuchi's metric if and only if they are uniformly relatively D-stable.
K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups …
Generalizes energy-momentum method for non-autonomous Hamiltonian systems.
problem Stability analysis of non-autonomous Hamiltonian systems with symmetries.
method Develops a new approach to relative equilibrium points and stability conditions for non-autonomous systems.
result Conditions ensuring stability of relative equilibrium points in non-autonomous Hamiltonian systems.
In this paper, we discuss the relative K-stability and the modified K-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative K-stability and the properness of modified K-energy. In …
Paper solves PDEs for optimal investment strategies in volatile markets.
problem Finding optimal investment strategies in volatile markets.
method Numerical methods using time-changed Bessel bridges.
result Solves PDEs for relative arbitrage opportunities in volatility-stabilized markets.
The paper improves asymptotic polybalanced kernels for extremal Kaehler metrics.
problem Stability and metrics on algebraic manifolds.
method Asymptotic polybalanced kernels associated to extremal Kaehler metrics.
result Stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds.
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.
Proves stability pulls back under proper actions, with applications to mapping class groups and free groups.
problem Stability of subgroups in mapping class groups and free groups.
method Proves stability pulls back under proper actions on metric spaces.
result Stability of convex cocompact subgroups in mapping class groups and free groups.
reval package selects best clustering solutions via stability-based validation.
problem Challenges in determining best clustering solutions due to lack of validation methods.
method Stability-based relative clustering validation methods.
result Determines best clustering solutions that generalize to unseen data.
Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
Study quantized extremal Kähler metrics for algebro-geometric stability.
problem Stability of extremal Kähler metrics and manifolds.
method Quantized extremal Kähler metrics and equivariant Riemann-Roch theorem.
result Proves weak relative Chow polystability and K-semistability.
New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
Eigenvalue estimate for shrinkers in mean curvature flow.
problem Eigenvalue estimates on shrinkers for mean curvature flow.
method Generalized earlier work of Ding and Xin to noncompact cases.
result Eigenvalue estimate holds on every properly embedded shrinker.