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.
Paper examines stability of quadratic curvature functionals on product Einstein manifolds.
problem Understanding stability of critical points of quadratic curvature functionals on product Einstein manifolds.
method Analyzes Riemannian functionals defined by L2-norms of Ricci, scalar, Weyl, and Riemannian curvatures. result Product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.
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.
New CVIs generalize scalar curvature stability results.
problem Stability and rigidity of Riemannian invariants.
method Establishing stability and rigidity results for CVIs.
result Generalizing scalar curvature stability results.
Study on stability of curvature functionals on manifolds.
problem Stability of quadratic curvature functionals on manifolds.
method Analysis of stability at constant sectional curvature metrics.
result Stability properties of quadratic curvature functionals.
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.
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.
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 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.
The article uses surgery on mean curvature flow to study level set flow's regularity and stability.
problem Analyzing the regularity and stability of level set flow.
method Using mean curvature flow with surgery to derive estimates.
result Demonstrates stability of the plane under level set flow.
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.
Establishes geodesic stability for Kähler metrics, proving existence of constant scalar curvature.
problem Existence of constant scalar curvature Kähler metrics.
method Exploring metric geometry of Mabuchi geodesic rays and uniform convexity properties of Kähler metrics space.
result Essentially optimal form of Donaldson's geodesic stability conjecture proved.
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.
Uniqueness and stability of minimal submanifolds proved.
problem Uniqueness and stability of minimal submanifolds.
method Proved a strong stability condition on minimal submanifolds.
result Existence and convergence of mean curvature flow for minimal submanifolds.
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.
Study shows curvature stability under smooth metric convergence.
problem Stability of nonnegative isotropic curvature under metric deformations.
method Introduced method by R. Bamler to study scalar curvature behavior.
result Proved that if metrics converge in C0 norm, resulting metric has isotropic curvature bounded from below.
Stability theorem for axisymmetric manifolds with nonnegative scalar curvature.
problem Stability of positive mass theorem for specific types of manifolds.
method Proved stability in W1,p sense with technical assumptions. result Derived estimates for volumes, areas, and distances.
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.
Study on stability of Positive Mass Theorem using Inverse Mean Curvature Flow.
problem Stability of Positive Mass Theorem in foliated regions with positive scalar curvature.
method Analyzes sequences of regions foliated by solutions to Inverse Mean Curvature Flow, focusing on convergence to flat annuli under specific conditions.
result Convergence of foliated regions to flat annuli under certain conditions, leading to stability of Positive Mass Theorem.
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 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.
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. Study stable maximal hypersurfaces in various spacetimes.
problem Characterize stability of maximal hypersurfaces in Lorentzian spacetimes.
method Analyze hypersurfaces in spacetimes with curvature assumptions, proving stability conditions and rigidity results.
result Characterize stability in spaces with constant sectional curvature and sufficient conditions for stability in general spacetimes.
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. 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. 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 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 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.
The paper proves stability of curvature bounds in geometric analysis.
problem Stability of local Riemannian Ricci curvature bounds under convergence.
method Gromov-Hausdorff convergence, Lagrangian approach, heat flow, weak gradients, Evolution Variational Inequality.
result Almost everywhere existence of Euclidean weak tangents.
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.