New manifolds with negative curvature limit to one with negative curvature.
problem Preserving nonnegative scalar curvature under intrinsic flat convergence.
method Constructing sequences of manifolds with positive scalar curvature.
result Intrinsic flat limit of manifolds with negative scalar curvature.
Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
problem Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
method Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations using specific curvature conditions.
result Explicit construction of Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
New compact K-E manifolds with negative curvature found.
problem Constructing compact Kähler-Einstein manifolds with negative curvature.
method Created compact Kähler-Einstein manifolds of dimension n with negative sectional curvature.
result Found compact Kähler-Einstein manifolds of negative curvature not covered by the ball.
Extending results about positive scalar curvature to non-negative curvature.
problem Extending results about positive scalar curvature to non-negative curvature.
method Using index theory to generalize results.
result Explicit generalizations of classical results about positive scalar curvature metrics to non-negative curvature metrics.
The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.
problem Understanding the ampleness of canonical line bundles for Kähler manifolds with specific curvature properties.
method Introducing a new notion of almost quasi-negative holomorphic sectional curvature and extending the theorem to this setting.
result The theorem is extended to compact Kähler manifolds with almost quasi-negative holomorphic sectional curvature, and a gap-type theorem is derived.
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
Study shows Kähler-Ricci flow on manifolds with negative curvature converges to a Kähler-Einstein metric.
problem Analyzing the behavior of Kähler-Ricci flow on manifolds with negative holomorphic curvature.
method Investigates the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature.
result The flow exists for all time and converges to a Kähler-Einstein metric of negative scalar curvature.
New metric with negative curvature found near positive-curvature spaces.
problem Understanding spaces with positive curvature bounds.
method Analyzing metrics in relation to curvature bounds.
result Near any point in a positive-curvature space, there exists a metric with negative upper curvature.
The study examines symmetries in spaces with positive or non-negative curvature.
problem Understanding symmetries in spaces with curvature constraints.
method Survey of existing results for Riemannian manifolds with specified curvature properties and symmetries.
result Results on symmetries in spaces with curvature bounds.
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
The paper proves complete conformal metrics with negative scalar curvatures have negative Ricci curvatures in convex domains.
problem Negativity of Ricci curvatures in complete conformal metrics.
method Analysis of Green's function and positive mass theorem.
result Complete conformal metrics with negative scalar curvatures have negative Ricci curvatures in convex domains.
Graphs with non-negative Ollivier curvature have constant bounded harmonic functions.
problem Liouville property for graphs with non-negative Ollivier curvature.
method Proving Liouville property and improving concentration results.
result Every bounded harmonic function on graphs with non-negative Ollivier curvature is constant.
Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.
problem Determining conditions for ample canonical line bundles in Hermitian manifolds.
method Hermitian curvature flow with specific curvature conditions.
result Canonical line bundle is ample under given curvature conditions.
Constructs 7-manifolds with SO(3) invariant non-negative curvature.
problem Finding exotic spheres with non-negative curvature.
method Six-parameter family of highly connected 7-manifolds constructed with invariant metrics.
result All exotic spheres in 7 dimensions have non-negative curvature.
Complete Finsler spaces with negative Ricci curvature are reversible.
problem Characterizing Finsler spaces with constant negative Ricci curvature.
method Utilizing projectively invariant pseudo-distance and Schwarzian derivative.
result Every connected complete Finsler space with constant negative Ricci scalar is reversible.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.
problem Characterize quaternionic bisectional curvature on quaternion-Kähler manifolds.
method Analyzing properties of quaternionic bisectional curvature on specific manifolds.
result Non-negative quaternionic bisectional curvature is only on quaternionic projective space.
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM3 manifolds of non-negative curvature. Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
Method constructs discrete surfaces with negative curvature.
problem Creating discrete surfaces with negative Gaussian curvature.
method Birkhoff decomposition and nonlinear d'Alembert formula.
result Simple algorithm for Birkhoff decomposition.
The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.
problem Proving properties of Sasakian manifolds with negative transverse holomorphic sectional curvature.
method Analyzing the curvature properties and applying the Wu-Yau theorem.
result Compact Sasakian manifolds with negative transverse holomorphic sectional curvature have negative transverse Ricci curvature.
Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.
problem Extending non-existence theorem of positive scalar curvature to product manifolds.
method Introduced intermediate curvature and studied rigidity conditions.
result Rigidity when intermediate curvature is non-negative in low dimensions.
Flat Higgs bundles restrict manifolds' curvature to be semi-negative.
problem Curvature restrictions on manifolds with Higgs bundles.
method Proving a semi-negative curvature property for manifolds with flat admissible Higgs bundles.
result Proved that curvature of manifolds with flat Higgs bundles is semi-negative.
Complete negative Kähler-Einstein metric found on Stein manifolds.
problem Existence of complete Kähler-Einstein metrics on Stein manifolds with negative curvature.
method Normalized Kähler-Ricci flow to deform metrics to complete negative Kähler-Einstein metric.
result Existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
The cross curvature flow preserves negative sectional curvature in 3-manifolds for all time.
problem Preserving negative sectional curvature in 3-manifolds under the cross curvature flow.
method Maximum principle to prove long-time existence of the flow.
result The flow exists for all time and converges to a hyperbolic metric.
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. 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.
Local Ricci flow under negative curvature conditions, with applications to metric space smoothing.
problem Finding solutions to Ricci flow under local negative curvature conditions.
method Local solution to Ricci flow equation with uniform existence time and bounded curvature.
result Local Ricci flow exists for a uniform time with bounded curvature, generalizing previous results.
The study examines the flexibility of entropies for negatively curved surfaces.
problem The study investigates the flexibility of topological and metric entropies for negatively curved surfaces.
method The authors compare different metrics on surfaces of negative curvature and analyze their topological and metric entropies.
result The study proves that the topological and metric entropies for metrics of negative curvature are flexible and only equal in the case of constant negative curvature.
Rigidity of negatively curved geodesic orbit spaces proven.
problem Proving rigidity of geodesic orbit Finsler spaces with non-positive curvature.
method Analyzing strictly negative flag curvature and non-compact Riemannian symmetric spaces of rank one.
result A geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
Study on Yamabe flow for negative scalar curvature.
problem Prescribed scalar curvature problem on compact manifolds.
method Yamabe flow with conditions on scalar curvature function.
result Long time existence and convergence of the flow.
The study proves the non-existence of certain Kähler metrics with specific curvature properties.
problem Non-existence of complete Kähler metrics with negatively pinched holomorphic sectional curvature.
method Construction of a Kähler metric with negatively pinched holomorphic sectional curvature and application of equivalence of invariant metrics.
result The dichotomy of completeness and non-existence of Kähler metrics with negatively pinched holomorphic sectional curvature.
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.
problem Proving the existence of complete Kähler metrics with negative holomorphic bisectional curvature in certain domains.
method Analyzing bounded domains in Cn with specific curvature properties. result Strictly pseudoconvex bounded domains and domains with squeezing function tending to 1 at boundary points admit complete Kähler metrics with negative holomorphic bisectional curvature everywhere.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.
problem Investigating properties of Kähler manifolds with specific curvature conditions.
method Conformal perturbation method.
result Established structure and splitting theorems for Kähler manifolds with non-negative mixed curvature.
Study on moduli spaces of non-negative curvature metrics on manifolds.
problem Understanding the topology of moduli spaces of non-negative curvature metrics.
method Construction of manifolds with specific curvature properties and analysis of their moduli spaces.
result First classes of manifolds with non-trivial rational homotopy, homology, and cohomology groups for moduli spaces of non-negative sectional curvature.
The study confirms a conjecture about Kähler manifolds with quasi-negative curvature.
problem Confirming a long-standing conjecture about Kähler manifolds with quasi-negative curvature.
method Introducing (ε,δ)--quasi-negativity and applying gap-type theorems. result Obtained gap-type theorems for ∫Xc1(KX)n>0 in terms of real bisectional curvature and weighted orthogonal Ricci curvature. Study on G2-structures with negative Ricci curvature on closed and noncompact manifolds.
problem Existence and properties of closed G2-structures with negative Ricci curvature. method Analyzing existence and nonexistence of closed G2-structures with negative Ricci curvature on closed and noncompact manifolds. result No closed manifold admits a closed G2-structure with negative Ricci curvature. For noncompact manifolds, restrictions on lengths of geodesics are found. The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.
problem Achieving metrics with negative Ricci curvature on closed Riemannian manifolds.
method Solving a fully nonlinear equation to conformally bend the manifold.
result Metrics of quasi-negative Ricci curvature are conformal to metrics with negative Ricci curvature.
The study examines non-negative curvature and conullity of curvature tensors.
problem Analyzing conullity conditions for curvature tensors and their compatibility with non-negative sectional curvature.
method Examined conullity two and three in specific dimensions, and used manifold properties to derive conditions.
result Finite volume manifolds with conullity 3 are locally products, and conditions are compatible with non-negative curvature only for specific manifold types.
Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.
problem Determining the minimum vertex degree for non-negative Lin-Lu-Yau curvature.
method Investigation of Ollivier-Ricci curvature and Lin-Lu-Yau modification on locally finite graphs.
result Lower bound on minimum vertex degree ensuring non-negative Lin-Lu-Yau curvature.
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.