Preserves positive intermediate curvature on manifolds.
problem Obstructs positive intermediate curvature on partial tori.
method Shows smooth interpolation of metrics with positive intermediate curvature.
result Proves non-existence of certain manifolds with positive intermediate 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.
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.
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.
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 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.
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 proves that certain manifolds with boundary cannot have metrics with positive intermediate curvatures.
problem Proving the nonexistence of metrics with positive intermediate curvatures on manifolds with boundary.
method Curvature obstruction theorems for manifolds with boundary.
result Topologically nontrivial compact manifolds with boundary cannot have metrics of positive m-intermediate curvature if the boundary is m-convex. 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…
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.
Sharp dimension constraints for positive intermediate curvature metrics are established.
problem Proving sharp dimension constraints for metrics with positive intermediate curvature.
method Constructing counterexamples and extending rigidity results.
result Sharp dimension constraints for positive intermediate curvature metrics are established.
New curvature concept shows certain manifolds can't have positive curvature.
problem Understanding manifolds that can't have positive curvature metrics.
method Introducing m-intermediate curvature and using stable weighted slicings. result Manifolds Nn=Mn−mimesTm do not admit positive m-intermediate curvature for n≤7. 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.
Extends metric properties over surgeries to higher codimensions.
problem Extending metrics of positive scalar curvature over surgeries in higher codimensions.
method Generalizes Gromov-Lawson, Schoen-Yau, and Walsh's construction to (p,n)-intermediate scalar curvature. result Shows infinitely many path components for certain metrics on (4n−1)-manifolds. 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.
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. Proves non-existence of metrics with positive curvature for certain connected sums.
problem Non-existence of metrics with positive curvature for specific connected sums.
method Using μ-bubbles, proves non-existence for various dimensions and manifolds.
result Connected sums do not admit metrics of positive scalar or intermediate curvature.
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. 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.
We establish metrics of positive 2nd-intermediate Ricci curvature, i.e. Ric2>0, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively …
The study constructs metrics with positive 2nd Ricci curvature on various manifolds.
problem Constructing metrics with positive 2nd Ricci curvature on closed manifolds.
method Generalization of the concept of fatness to ensure the existence of metrics with positive 2nd Ricci curvature on certain homogeneous bundles.
result Infinitely many examples of manifolds with positive 2nd Ricci curvature, including non-simply connected spaces.
We formulate extensions of Wilking's Jacobi field splitting theorem to uniformly positive sectional curvature and also to positive and nonnegative intermediate Ricci curvatures.
Study shows Gromov's Betti number bound fails for certain intermediate Ricci curvatures.
problem Gromov's Betti number bound for sectional curvature bounded below does not hold for intermediate Ricci curvatures.
method Established a surgery result for Riemannian metrics with Rick>0 and showed failure of Gromov's bound for specific ranges of k. result Gromov's Betti number bound fails for Rick>0 when ⌊n/2floor+2≤k≤n−1. 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.
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. Study proves rigidity of certain hypersurfaces in 5- and 6-manifolds.
problem Proving rigidity of stable minimal hypersurfaces in 5- and 6-manifolds.
method Nonnegative 3-intermediate Ricci curvature combined with uniformly positive k-triRic curvature.
result No complete noncompact stable minimal hypersurface in a closed 5-dimensional manifold with positive sectional curvature.
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 paper proves manifold splitting theorems with nonnegative intermediate curvature.
problem Proving rigidity results for manifolds with nonnegative intermediate curvatures.
method New recursion theorem for spectral intermediate curvatures and cylindrical splitting theorems.
result Smooth metrics with uniformly positive intermediate curvature constructed.
The study proves stable minimal immersions in positively curved manifolds are totally geodesic.
problem Proving stable minimal immersions in positively curved manifolds are totally geodesic.
method Formulating stable Bernstein type theorems in certain positively curved ambient manifolds.
result Proves stable minimal immersions in positively curved manifolds are totally geodesic.
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…
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive curvature.
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.
New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.
problem Proving rigidity of minimal hypersurfaces in curved 4-manifolds.
method Combining nonnegative 2-intermediate Ricci curvature and strict positivity of scalar curvature, extending Chodosh-Li-Stryker method.
result Rigidity of two-sided free boundary stable minimal hypersurfaces in 4-manifolds with bounded geometry and weakly convex boundary.
We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues…
Paper introduces p-Laplace equations for curvature in conformal geometry.
problem Study of geometry and topology of manifolds.
method Introducing p-Laplace equations for intermediate Schouten curvature.
result Nonnegative intermediate Schouten curvature leads to estimates on Hausdorff dimension of singular sets and vanishing of homotopy groups.
In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we …
The space of metrics with positive Ricci curvature on spheres has special algebraic structures.
problem Understanding the space of metrics with positive Ricci curvature on spheres.
method Using H-space and loop space structures, and operad theory.
result The space of metrics with positive Ricci curvature on spheres is homotopy equivalent to an n-fold loop space.
Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.
problem Existence and non-existence of positive solutions for the Lane-Emden equation on Riemannian models.
method Analysis of the subcritical Lane-Emden equation on various Riemannian manifolds with polynomial volume growth.
result Subcritical regime divides into three ranges with distinct existence and non-existence phenomena.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
Paper proves inequality for p-Laplacian eigenvalues on curved spaces.
problem Eigenvalue inequalities for p-Laplacian on curved manifolds.
method Robin boundary conditions, lower Ricci bounds, positive asymptotic volume ratio.
result Bossel-Daners inequality extends to compact submanifolds.
Optimizes transport on submanifolds for curvature inequalities.
problem Proving Michael-Simon-Sobolev inequalities in manifolds with intermediate Ricci curvature bounds.
method Generalizes optimal transport theory to submanifolds and applies to curvature inequalities.
result Proves a variant of the Michael-Simon-Sobolev inequality in manifolds with nonnegative intermediate Ricci curvatures.
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
Log-Sobolev inequality proven for submanifolds in specific types of manifolds.
problem Proving Log-Sobolev inequality for submanifolds in asymptotic non-negative intermediate Ricci curvature manifolds.
method Extending previous results, proving inequality for submanifolds in specific types of manifolds.
result Sharp Log-Sobolev inequality proven for submanifolds in complete non-compact Riemannian manifolds with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth.
Solves curvature equations using parabolic flows in various spaces.
problem Finding strictly convex, spacelike solutions to curvature equations.
method Employing curvature flows without global terms in Riemannian and Lorentzian spaces.
result Solves a broad class of elliptic prescribed curvature equations.
Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
problem Proving Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
method Using the Alexandrov-Bakelman-Pucci method to prove Michael-Simon type inequalities.
result Extends existing inequalities to the k-Ricci curvature setting and provides isoperimetric inequalities. Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
Proves rigidity of stable free boundary hypersurfaces in 5-manifolds.
problem Stability and rigidity of free boundary hypersurfaces in 5-manifolds.
method Combining k-tri-Ricci curvature and 3-intermediate Ricci curvature. result Improves rigidity result to 5-dimensions and extends to free boundary case.
New technique identifies submanifolds in symmetric spaces based on Ricci curvature.
problem Identifying submanifolds in symmetric spaces of compact type.
method Computing k-positive Ricci curvature and using it to determine submanifold connectivity. result Codimension ranges for submanifolds with specific conditions.