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.
Stability of capillary hypersurfaces with higher order mean curvature.
problem Stability of capillary hypersurfaces with constant higher order mean curvature.
method Generalization of classical stability theory for capillary hypersurfaces.
result Results on stability for capillary hypersurfaces with higher order mean curvature.
New rigidity results for scalar curvature with stabilized conditions.
problem Establishing rigidity for scalar curvature with stabilized conditions.
method Construction of foliations and development of a monotone quantity using Ricci flow and heat equation.
result Generalized classical scalar curvature rigidity results to the \(T^{
times}\)-stabilized setting.
The study shows stability of neckpinch singularities in mean curvature flows.
problem Stability of neckpinch singularities in mean curvature flows.
method Analysis of mean curvature flow and perturbations.
result Stability of neckpinch singularities in mean curvature flows.
Study on stability of constant mean curvature hypersurfaces in Riemannian manifolds.
problem Stability of constant higher mean curvature hypersurfaces in Riemannian manifolds.
method Introduced a new notion of stability and used two stability operators to relate it to the first eigenvalues. Applied to Space Forms and proved non-stability for certain hypersurfaces.
result Embedded rotational spheres with constant k-mean curvature in HnxR or SnxR are not stable.
Survey on scalar curvature stability and related questions.
problem Understanding scalar curvature stability and rigidity phenomena.
method Survey and discussion of existing tools and questions.
result Survey of known results and open questions in scalar curvature stability.
Study adds scalar curvatures of mapped manifolds to Riemannian products.
problem Additivity of scalar curvatures in Riemannian products.
method Stabilized scalar curvatures of mapped manifolds.
result Proves additivity in some cases.
We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
Study on stability of hyperkähler flow in 4-manifolds.
problem Stability of hyperkähler flow in 4-manifolds.
method Extending results from mean curvature flow for minimal surfaces to hyperkähler flow.
result Obtained a dynamic stability theorem for hyperkähler flow.
Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
Paper uses Ricci curvature to measure and forecast China's stock market stability.
problem Measuring and predicting systemic stability of China's stock market.
method Geometric measure derived from discrete Ricci curvature applied to financial networks.
result Ricci curvature effectively captures market stability and predicts future trends.
The paper examines the stability of Minkowski inequality for nearly spherical domains.
problem Stability of Minkowski inequality for nearly spherical domains.
method Analyzes stability inequalities for C1 perturbations of a ball and axially symmetric perturbations. result Established stability inequalities for curvature integrals of nearly spherical domains.
Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.
problem Preserving symplectic forms under deformations on Calabi-Yau manifolds.
method Dynamical stability of symplectic curvature flow.
result Any small symplectic deformation of a Kähler form remains Kähler on a compact Calabi-Yau manifold.
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. 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.
Study on Einstein manifolds linking stability and rigidity.
problem Einstein manifold rigidity and stability.
method Review of linear and dynamical stability, scalar curvature rigidity.
result Relation between stability and rigidity of Einstein manifolds.
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
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. Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
The study examines stability of Sobolev inequalities on manifolds with Ricci bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Combines techniques from smooth and non-smooth geometry, focusing on direct strategies.
result Effective methods revealed for stability of Sobolev inequalities on manifolds with non-negative Ricci curvature and Euclidean volume growth.
Stability of positive mass theorem proven under Ricci curvature bounds.
problem Stability of positive mass theorem under Ricci curvature lower bounds.
method Harmonic level set approach combined with techniques from almost splitting theorem.
result Proves Gromov-Hausdorff stability of positive mass theorem.
The study proves stability of various graphical translators in mean curvature flow.
problem Stability of graphical translators in mean curvature flow.
method Existence of longtime solution to mean curvature flow, dynamical stability results for various graphical translators.
result Dynamical stability of various types of graphical translators.
Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.
problem Stability of generalized Lagrangian mean curvature flow in cotangent bundle.
method New derivative estimates to weaken initial conditions and remove curvature constraints.
result Stability of flow near special Lagrangian submanifolds in cotangent bundle.
The paper proves stability and convergence of minimal networks under curvature motion.
problem Stability and convergence of minimal networks under curvature motion.
method Proved Lojasiewicz-Simon gradient inequalities for minimal networks.
result Motion by curvature starting from networks close to minimal ones exists for all times and smoothly converges.
Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in R7. The study examines rigidity and stability of gradient estimates on surfaces and manifolds.
problem Rigidity and stability of gradient estimates for positive harmonic functions and solutions to heat equations.
method Sharp gradient estimates for positive harmonic functions and solutions to heat equations on surfaces and manifolds with nonnegative curvature.
result Obtained rigidity and stability results for gradient estimates.
Minimal Lagrangians in certain curved spaces are stable under specific flows.
problem Stability of minimal Lagrangians in Kähler-Einstein manifolds of non-positive curvature.
method Proved stability under Lagrangian mean curvature flow.
result Equivalence between linear and dynamical stability for C1-close Lagrangians. Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections implies exponential growth of perturbations and hence suggests instability, whi…
The paper examines stability of Yamabe boundary problem under perturbations.
problem Stability of Yamabe boundary problem under perturbations of mean curvature and scalar curvature.
method Analyzes stability of the Yamabe boundary problem with respect to perturbations of mean curvature and scalar curvature.
result The stability of the Yamabe boundary problem is proven under perturbations from below, but not from above.
Sharp stability estimate for tensor tomography in non-positive curvature.
problem Stability estimate for tensor tomography on manifolds with non-positive curvature.
method Pestov identity with localized frequency boundary term.
result Stability estimate of the form L2↦HT1/2. Introduces a universal Bochner formula for scalar curvature.
problem None explicitly stated; focuses on a new formula.
method Introduces a universal Bochner formula.
result Contains special cases like stability inequality and Schrödinger-Lichnerowicz-type formula.
The paper examines the stability of a specific flow on complex manifolds.
problem Stability of line bundle mean curvature flow on complex manifolds.
method Analyzes the convergence of the line bundle mean curvature flow to a deformed Hermitian-Yang-Mills metric.
result The flow converges exponentially to the deformed Hermitian-Yang-Mills metric in the C∞ sense. The study examines stability of triharmonic hypersurfaces in space forms.
problem Stability of triharmonic hypersurfaces in space forms.
method Derivation of general stability statements, focus on specific cases of constant mean curvature in Euclidean and hyperbolic spaces, and analysis of small proper triharmonic hyperspheres and Clifford tori.
result Triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations, while in hyperbolic space they are stable.
Study on stability of mean curvature flow in hyperbolic space.
problem Stability of volume preserving mean curvature flow in hyperbolic space.
method Analysis of initial conditions and flow behavior in hyperbolic space.
result The flow converges exponentially to an umbilical sphere under certain conditions.
Study on deformation of weighted scalar curvature, proving geometric results and stability.
problem Deformation of weighted scalar curvature and related geometric properties.
method Linearization of weighted scalar curvature, studying kernel of formal adjoint.
result Definition and study of weighted vacuum static spaces, stability results on flat spaces.
Study proves stability and uniqueness for a specific type of flow.
problem Volume-preserving mean curvature flow stability and uniqueness.
method New gradient flow calibrations for volume preservation, stability estimate in distributional solutions.
result Strong solutions are calibrated and stable under certain conditions.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
New examples of manifolds with lower scalar curvature bounds and submanifold collapse.
problem Stability of scalar curvature rigidity phenomena.
method Constructing Riemannian manifolds with specific curvature and collapse properties.
result Examples demonstrating stability and rigidity of scalar curvature.
The round sphere is stable among spin manifolds with a specific scalar curvature bound.
problem Stability of the round sphere among spin manifolds with scalar curvature below a certain threshold.
method Showed that if the scalar curvature is bounded from below by n(n−1)−ε, the manifold is C0-close to a finite number of spheres outside a small bad set. result The spherical stability problem is completely solved.
The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.
problem Proving curvature bounds in non-smooth spaces.
method Extending results from smooth Riemannian manifolds to non-smooth RCD spaces.
result Stability of mean curvature bounds under uniform convergence.
The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.
problem Geometric stability of the positive mass theorem and related conjectures.
method Analysis of harmonic maps to flat model spaces under integral curvature bounds.
result Upgrading harmonic maps to diffeomorphisms when conditions are met, proving quantitative closeness to model spaces.
The study examines stability of metric measure spaces with integral Ricci curvature bounds.
problem Stability and compactness of metric measure spaces with integral Ricci curvature bounds.
method Proves convergence to metric measure spaces satisfying CD(K,n) condition under certain curvature bounds. result Proves convergence of sequences of Riemannian manifolds to metric measure spaces satisfying CD(K,n) condition. Proves a conjecture about manifolds and scalar curvature.
problem Determining when a manifold admits a positive scalar curvature metric.
method Uses a geometric bound to measure discrepancies between vector fields.
result Proves the conjecture in codimension two.
Study Kähler metrics on complex tori with almost non-negative scalar curvature.
problem Stability of Kähler metrics on complex tori.
method Proved convergence of non-collapsing subsequence of Kähler metrics to flat torus.
result Kähler metrics with almost non-negative scalar curvature on complex tori converge to flat torus.
New stability concept for Poisson structures leads to constant curvature metrics.
problem Finding constant scalar curvature metrics in generalized Kähler geometry.
method Introducing Poisson K-stability and using infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
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.