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.
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…
If π:M→B is a Riemannian Submersion and M has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…
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 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.
New rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.
problem Understanding the structure of manifolds with maximal symmetry rank and positive intermediate Ricci curvature.
method Recovering stronger topological rigidity results using higher intermediate Ricci curvatures and nontrivial fundamental groups.
result Stronger topological rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.
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
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.
Extends Bochner's theorem to include small positive Ricci curvature.
problem Classical Bochner theorem limitations.
method Extensions with small positive Ricci curvature.
result Isometry group is finite for small positive curvature.
Surveying problems with positive curvature forms, focusing on Ricci flow.
problem Problems with Riemannian manifolds and positive curvature forms.
method Explains how recent Ricci flow theory can solve these problems.
result Ricci flow is well-suited for solving problems involving positive 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.
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.
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.
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.
In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature.
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
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 …
Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one positive Ricci curvature is equivalent to positive Ricci curvature and n-…
5 minimal tori found in 3-spheres with positive Ricci curvature.
problem Existence of at least 5 embedded minimal tori in 3-spheres with positive Ricci curvature.
method Combination of min-max theory and mean curvature flow heuristics.
result Confirms B. White's conjecture for positive Ricci curvature.
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.
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.
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.
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.
Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.
problem Rigidity of harmonic maps between manifolds with curvature constraints.
method Proves an extension of Eells-Sampson theorem under positive sectional curvature upper bounds.
result Recover Hamilton's rigidity result for positive Ricci curvature.
Maximal diameter theorem for graphs with positive Ricci curvature.
problem Diameter comparison in directed graphs with positive Ricci curvature.
method Introduced a Lin-Lu-Yau type Ricci curvature for directed graphs and investigated rigidity properties for the equality case.
result Concluded a maximal diameter theorem of Cheng type.
Optimal volume limit found for Kähler manifolds with positive Ricci curvature.
problem Bounding the volume of Kähler manifolds with positive Ricci curvature.
method Using δ-invariants and Newton--Okounkov bodies.
result Derive the optimal volume upper bound and new characterization of the complex projective space.
In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.
Study on positive scalar curvature and its impact on Ricci limit spaces.
problem The influence of uniformly positive scalar curvature on Ricci limit spaces.
method Investigates uniformly positive scalar curvature on non-collapsed Ricci limit spaces.
result Proves a limit space splits at most n-2 lines or R-factors.
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 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.
The paper studies curvature properties under a specific type of flow on spaces with conical singularities.
problem Preserving curvature properties (Ricci curvature and scalar curvature) under a flow with conical singularities.
method Ricci de Turck flow, preserving conical structure, additional assumptions for scalar curvature positivity.
result Positivity of scalar curvature is preserved under the flow with additional assumptions.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
The paper examines gradient Ricci solitons on orbifolds and proves their rigidity properties.
problem The rigidity of positively curved gradient Ricci solitons on orbifolds.
method Analyzes scalar curvature, uses nonnegative curvature operator, κ-noncollapsed condition, and asymptotic quotient cylindrical properties.
result Steady gradient Ricci solitons on orbifolds with positive curvature are rigid and must be quotients of the Bryant soliton.
We use a local argument to prove if an r-dimensional torus acts isometrically and effectively on a connected n-dimensional manifold which has positive kth-intermediate Ricci curvature at some point, then r≤⌊2n+k⌋. This symmetry rank bound generalizes those established by Gr…
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.
New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.
problem Constructing manifolds with positive Ricci curvature and non-abelian fundamental groups.
method Constructing specific 9-dimensional manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
result Examples of manifolds with positive Ricci curvature and non-uniformly virtually abelian fundamental groups.
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.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
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…
New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.
problem Proving Sobolev inequalities on Kähler manifolds.
method Using a classical Bidaut-Véron and Véron approach.
result Proved new Sobolev inequalities on compact Kähler manifolds with positive Ricci curvature.
The study finds large Betti numbers in minimal hypersurfaces with positive Ricci curvature.
problem Minimal hypersurfaces with large Betti numbers in manifolds with positive Ricci curvature.
method Constructing sequences of manifolds with embedded minimal hypersurfaces.
result Minimal hypersurfaces have unbounded first Betti numbers.
Riemannian submersions can preserve positive intermediate Ricci curvature, but not necessarily.
problem Understanding the conditions under which Riemannian submersions preserve positive intermediate Ricci curvature.
method Analyzing the Gray--O'Neill Horizontal curvature equation and constructing perturbations of metrics.
result Riemannian submersions that do not preserve positive Ricci curvature are dense in the C1-topology. We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.
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.
Sharp gradient estimates for positive Ricci curvature manifolds.
problem Understanding geometric properties of manifolds with positive Ricci curvature.
method Proving sharp gradient estimates and monotonicity formulae.
result Sharp gradient estimates and monotonicity formulae for positive Ricci curvature manifolds.
Study manifolds with positive intermediate Ricci curvature and large symmetry rank.
problem Closed, simply connected manifolds with positive 2nd-intermediate Ricci curvature and large symmetry rank.
method New tools for studying isometric actions on closed manifolds with positive kth-intermediate Ricci curvature, including isotropy rank lemma, symmetry rank bound, and connectedness principle.
result Even dimensional manifolds with large symmetry rank must have trivial odd degree integral cohomology and are either spheres or complex projective spaces.
Riemannian metrics of positive Ricci curvature were constructed on certain moment-angle manifolds.