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.
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.
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.
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.
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.
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.
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.
Ricci flow can change metrics with intermediate curvatures.
problem Ricci flow's preservation of curvature conditions.
method Construction of specific homogeneous spaces and Ricci flow evolution.
result Ricci flow can alter metrics with intermediate curvatures, not preserving certain curvature conditions.
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
The Ricci flow preserves positivity on Stiefel manifolds.
problem Preserving positivity of Ricci curvature on Stiefel manifolds.
method Normalized Ricci flow on Stiefel manifolds.
result Normalized Ricci flow evolves metrics with mixed Ricci curvature into positive ones.
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. 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 shows metrics on certain manifolds lose positive curvature under Ricci flow.
problem Understanding the dynamics of positively curved metrics on specific manifolds.
method Analysis of invariant metrics on SU(3)/T2 and SU(m+2p)/S(U(m)imesU(p)imesU(p)) under homogeneous Ricci flow. result Metrics lose positive intermediate Ricci curvature under Ricci flow for certain dimensions.
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.
We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expa…
This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces SU(3)/Tmax, Sp(3)/Sp(1)×Sp(1)×Sp(1), and F4/Spin(8). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curv…
Establishes metrics with positive 2nd intermediate Ricci curvature on products of curved spaces.
problem Examines the limitations of positive curvature metrics on product spaces.
method Uses examples to show Ric2>0 does not imply positive curvature for products of spaces. result The Ric2>0 class of manifolds is distinct from positively curved manifolds. It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on …
In this paper we construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a…
We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E …
We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature.
Paper proves metrics of positive Ricci curvature on fiber bundles.
problem Existence of metrics with positive Ricci curvature on fiber bundles.
method Direct and simple proof, generalizing Nash and Poor's results.
result New examples of manifolds with positive Ricci curvature constructed.
Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.
problem Finding metrics with specific curvature properties on Brieskorn quotients.
method Analyzing moduli spaces of metrics with nonnegative sectional or positive Ricci curvature.
result Moduli spaces have infinitely many path components for both nonnegative sectional and positive Ricci curvature.
We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.
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.
The article shows how to create metrics with positive Ricci curvature on twisted suspensions.
problem Creating metrics with positive Ricci curvature on complex manifolds.
method Using twisted suspensions and Riemannian metrics.
result Maximal symmetry rank of positive Ricci curvature manifolds is (n-2) in all dimensions n≥4.
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.
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced hh-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…
Minimal surfaces in lens spaces identified with specific counts.
problem Identifying minimal surfaces in lens spaces.
method Variant multiplicity one theorem for Simon-Smith min-max theory under equivariant settings.
result Existence of distinct minimal surfaces in lens spaces.
We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonn…
Riemannian metrics of positive Ricci curvature were constructed on certain moment-angle manifolds.
We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some top…
This paper describes metrics with varying curvature properties in a specific manifold.
problem Characterizing regions in a manifold with specific curvature properties.
method Using a projected Ricci flow and studying the dynamics of regions in the manifold.
result Sign curvature maintenance and escaping in regions of the manifold.
The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.
problem Finding different geometries for odd-dimensional manifolds with positive Ricci curvature.
method Examining closed manifolds in odd dimensions n≥5. result Large classes of manifolds admit infinitely many different geometries of positive Ricci curvature.
The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.
problem Classifying and understanding moduli spaces of metrics with specific curvature properties on sphere bundles.
method Analyzing total spaces of S7-bundles over S8 and quotients of Milnor and Shimada spheres. result The moduli space of metrics has infinitely many path components.
We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k-fold connected sum of S^{2n-1} x S^{2n} admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for …
Extends Perelman's theorem to positive intermediate curvature conditions.
problem Positive intermediate curvature conditions and their implications.
method Generalization of Perelman's gluing theorem to positive intermediate curvature conditions.
result Observer moduli space can have non-trivial higher homotopy groups.
Curved metrics on Wallach spaces bounded by curves under flow.
problem Properties of positively curved Riemannian metrics on Wallach spaces.
method Normalized Ricci flow analysis on specific Wallach spaces.
result Set of metrics forms bounded curves asymptotically.
The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
problem Investigating volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
method Applying similar techniques to derive local rigidity theorems for strictly stable Ricci flat manifolds.
result Derives local rigidity theorems for strictly stable Ricci flat manifolds with respect to σ2-curvature.
In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…
We show that the Kähler-Ricci flow on a manifold with positive first Chern class converges to a Kähler-Einstein metric assuming positive bisectional curvature and certain stability conditions.
The paper finds many manifolds with intermediate Ricci curvature for small k.
problem Finding manifolds with intermediate Ricci curvature.
method Examining symmetric and normal homogeneous spaces, along with metric deformations of fat homogeneous bundles.
result Proves existence of infinitely many manifolds with Rick>0 for some k<n/2. For homogeneous metrics on the spaces of the title it is shown that the Ricci flow can move a metric of stricly positive sectional curvature to one with some negative sectional curvature and one of positive definite Ricci tensor to one with indefinite signature.
The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.
problem Preserving Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.
method Investigation of infinitely many generalized Wallach spaces (GWS) where Ricci curvature positivity is preserved.
result Infinite number of GWS where Ricci curvature positivity is preserved under the normalized Ricci flow.
The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…
We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.