The paper examines positivity properties of singular Hermitian metrics.
problem Investigating positivity for singular Hermitian metrics.
method Exploring Griffiths, ω-trace, and RC positivity.
result These positivity notions imply cohomology vanishing and rational connectedness.
The paper extends positivity results from vector bundles to Kobayashi positive ones.
problem Extending positivity results from vector bundles to Kobayashi positive ones.
method Using convexity of Kobayashi positive Finsler metrics and duality for convex Finsler metrics.
result The quotient and tensor product of Kobayashi positive vector bundles are also Kobayashi positive.
We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions,…
Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
Proves metrics with positive intermediate Ricci curvature on complex manifolds.
problem Establishing metrics with positive intermediate Ricci curvature on complex manifolds.
method Canonical variation and surgery techniques.
result Existence of metrics with positive intermediate Ricci curvature on various examples.
Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.
Generalizes surgery theorem for positive Ricci curvature metrics.
problem Preserving positive Ricci curvature under surgery.
method Gluing a sphere bundle with a core metric.
result Constructs new metrics of positive Ricci curvature.
New metrics found for 6k-dimensional manifolds with positive Ricci curvature.
problem Finding metrics of positive Ricci curvature on simply-connected manifolds.
method Using labeled bipartite graphs to describe certain manifolds and constructing metrics of positive Ricci curvature.
result Many new examples of 6k-dimensional manifolds with positive Ricci curvature.
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.
problem Rigidity of non-positively curved homogeneous Finsler metrics.
method Analyzes and proves rigidity results for specific Finsler metrics with non-positive flag curvature.
result Homogeneous Finsler spaces with non-positive flag curvature and isotropic S-curvature are either Riemannian or locally Minkowskian.
Curvature of Bazaikin spaces fully characterized, with unique quasi-positive metric.
problem Characterizing sectional curvature of Bazaikin spaces.
method Completely characterized the sectional curvature of all 13-dimensional Bazaikin spaces.
result Unique quasi-positively curved Riemannian metric on Bazaikin spaces, with a unique almost positively curved but not positively curved space.
The study proves that certain manifolds can have metrics with specific volume growth.
problem Determining if manifolds with positive scalar curvature can have metrics with a given volume growth.
method Using Gromov-Lawson and Grimaldi-Pansu constructions, the study proves the existence of metrics with the desired volume growth on specific manifolds.
result The study positively answers the question for manifolds that are infinite connected sums of closed manifolds with positive scalar curvature.
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on Sn for n≥4. result Any conformal metric to the round metric on Sn for n≥4 can be a limit of metrics with positive scalar curvature. Constructs uniformly positive scalar curvature metrics on open manifolds
problem Finding uniformly positive scalar curvature metrics on open manifolds
method Using Morse functions and exhaustion
result Proving the existence of uniformly positive scalar curvature metrics
Study preserves metrics with positive Bakry-Émry Ricci curvature via surgery.
problem Preserving metrics with positive Bakry-Émry Ricci curvature after surgery.
method Established theorems for connected sums and surgeries along higher-dimensional spheres.
result Local surgery results for positive Bakry-Émry Ricci curvature.
Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.
problem Existence of invariant metrics with positive intermediate Ricci curvature on low-dimensional cohomogeneity one manifolds.
method Construction of invariant metrics with positive intermediate Ricci curvature on specific manifolds.
result Invariant metrics with positive 4th-intermediate Ricci curvature exist but not for 3rd-intermediate Ricci curvature on certain manifolds.
The study finds non-trivial elements in moduli spaces of curvature metrics.
problem Identifying non-trivial elements in moduli spaces of curvature metrics.
method Constructing elements in homotopy groups of moduli spaces of positive curvature metrics.
result Non-trivial elements found in moduli spaces of positive curvature metrics.
Proves positive energy conjecture for a specific metric class.
problem Proving the positive energy conjecture for a class of AHM metrics.
method Analyzes asymptotically Horowitz-Myers (AHM) metrics on R2imesTn−2. result Generalizes previous results on positive energy conjecture.
We prove that the connected sums CP_2 # CP_2 and CP_2 # CP_2 # CP_2 admit self-dual metrics with positive Ricci curvature. Moreover, every self-dual metric of positive scalar curvature on CP_2 # CP_2 is conformal to a metric with positive Ricci curvature.
Solves Lempert's question on Nakano semi-positivity preservation.
problem Preserving Nakano semi-positivity under limits of metrics.
method Establishes L2 extension theorem and uses multiplier submodule sheaves. result Affirmative solution to Lempert's question on Nakano semi-positivity.
We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space Wloc2,n/2(M). We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive mass theorem may be proved by our methods.
The abstract discusses metrics with positive biorthogonal curvature on 5-manifolds.
problem Finding metrics with positive biorthogonal curvature on simply connected 5-manifolds.
method Using conformal deformation of Wilking's metric and results from Smale.
result Every closed simply connected 5-manifold admits a metric with strictly positive average sectional curvatures of orthogonal 2-planes.
Ricci flow can change metrics with positive curvature to those without.
problem Preserving positive sectional curvature under Ricci flow in specific dimensions.
method Examined SU3- and SU5-invariant metrics on Aloff-Wallach spaces and Berger space. result Found metrics with positive sectional curvature that evolve to non-positively curved metrics under Ricci flow.
We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, i…
Study on spaces of metrics with intermediate curvature bounds.
problem Understanding spaces of metrics with lower bounds on intermediate curvatures.
method Analyzing spaces of Riemannian metrics with specific curvature bounds on high-dimensional Spin-manifolds.
result Spaces of metrics with positive p-curvature and k-positive Ricci curvature have non-trivial homotopy groups.
The study finds positive Einstein metrics on complex manifolds and spheres.
problem Existence of positive Einstein metrics on complex manifolds and spheres.
method Investigation of cohomogeneity one metrics and use of known Einstein metrics.
result Existence of positive Einstein metrics on S4m+4 and S8. The study confirms positivity of Q-curvatures for specific conformal metrics.
problem Analyzing positivity of Q-curvatures for conformal metrics with certain properties.
method Examined a specific form of conformal metric and used properties of Q-curvatures to prove positivity.
result Lower order Q-curvatures are also positive under certain conditions.
Study torsion obstructions to positive scalar curvature on manifolds.
problem Obstructions to positive scalar curvature metrics on manifolds.
method Analysis of torsion classes in integral homology.
result Construction of manifolds without positive scalar curvature metrics but whose Cartesian products do.
Study shows infinitely many path components for positive Ricci curvature metrics on certain spin manifolds.
problem Understanding the space of metrics with positive Ricci curvature on spin manifolds.
method Analyzing spin manifolds of dimension 4k-1 for k ≥ 2, focusing on metrics with a specific property.
result The space of metrics has infinitely many path components.
The paper sets new bounds on metrics with positive scalar curvature.
problem Classifying metrics with positive scalar curvature.
method Lower bounds on the rank of the group of psc metrics over M.
result Lower bounds on the rank of psc metrics up to bordism.
Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.
problem Contractibility of spaces of invariant positive scalar curvature metrics.
method Combining equivariant Morse theory with conformal deformations and local flexibility properties.
result Spaces of invariant positive scalar curvature metrics are contractible.
Study deforms Hermitian metrics with positive curvature.
problem Deforming Hermitian metrics with positive curvature.
method Adapted conformal perturbation method to Hermitian setting.
result Hermitian metrics with quasi-positive curvature can be deformed to positive curvature.
New examples show positive scalar curvature metrics on manifolds with boundary that cannot be extended.
problem Positive scalar curvature metrics on manifolds with boundary that cannot be extended.
method Analytic techniques related to the prescribed scalar curvature problem in conformal geometry.
result Obstruction to positivity of conformal Laplacians given by a real-valued ξ-invariant.
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
Ricci flow deforms metrics with positive curvature to include negative curvature.
problem Preserving positive sectional curvature under Ricci flow in dimension four.
method Evolved cohomogeneity one metrics on S4 and CP2 via Ricci flow. result Metrics with positive sectional curvature lose this property under Ricci flow.
We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one s…
The study allows for connected sums in manifolds with positive intermediate Ricci curvature.
problem Performing connected sums in manifolds with positive intermediate Ricci curvature.
method Introducing and utilizing k-core metrics to show the possibility of connected sums. result Connected sums are possible under certain conditions involving k-core metrics. Study on metrics with positive scalar curvature on manifolds with singularities.
problem Understanding metrics with positive scalar curvature on manifolds with singularities.
method Proving homotopy invariance of the space of metrics with positive scalar curvature on manifolds with fibred singularities.
result Proved that the space of metrics with positive scalar curvature is homotopy invariant under certain surgeries.
We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations of some classical results concerning moduli spaces of positive scalar curvature m…
The study distinguishes components of moduli spaces of Ricci-positive metrics on 5-manifolds.
problem Distinguishing components of moduli spaces of Ricci-positive metrics.
method Using η invariants of spinc Dirac operators to classify and find infinitely many non-diffeomorphic 5-manifolds. result Infinitely many non-diffeomorphic 5-manifolds with infinitely many components in their moduli spaces of Ricci-positive metrics.
Study on moduli space of metrics with positive scalar curvature.
problem Topology of moduli space of metrics with positive scalar curvature.
method Analyzes the moduli space of concordances of metrics with positive scalar curvature.
result Shows non-trivial homotopy groups in the moduli space.
Unified positive mass theorem and Dirac operator study on weighted manifolds.
problem Establishing a unified positive mass theorem for weighted manifolds and smooth metric measure spaces.
method Analyzing Dirac operators on warped product manifolds and applying results to the positive mass theorem.
result Equivalence of weighted positive mass theorem to usual positive mass theorem.
The study connects manifold topology to metrics with positive intermediate curvature.
problem Understanding the relationship between manifold topology and metrics with positive intermediate curvature.
method Formulated a conjecture and proved it for specific dimensions and conditions.
result Closed, aspherical 6-manifolds cannot admit metrics with positive 4-intermediate curvature.
A Finsler space (M,F) is called flag-wise positively curved, if for any x∈M and any tangent plane P⊂TxM, we can find a nonzero vector y∈P, such that the flag curvature KF(x,y,P)>0. Though compact positively curved spaces are very rare in both Riemannian and Finsler g…
The study preserves positive Ricci curvature on connected sums of fibre bundles.
problem Preserving positive Ricci curvature on connected sums of fibre bundles.
method Lifting core metrics along general fibre bundles and applying to specific spaces.
result All classes in the torsion-free oriented bordism ring can be represented by connected manifolds of positive Ricci curvature.
Four minimal spheres found in sphere with special metric.
problem Existence of minimal spheres in spheres with specific metrics.
method Simon-Smith min-max theory for multiplicity one theorem.
result At least four embedded minimal 2-spheres proven.
Ricci flow preserves positive sectional curvature on homogeneous spheres
problem Classification of positively curved metrics on homogeneous spaces
method Proving Ricci flow preserves positive sectional curvature on homogeneous spheres
result Completes classification of positively curved metrics on homogeneous spaces