In this article, we generalize the classical Bochner-Weitzenböck theorem for manifolds satisfying an integral pinching on the curvature. We obtain the vanishing of Betti numbers under integral pinching assumptions on the curvature, and characterize the equality case. In particular, we reprove and extend to higher degre…
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48 results for “pinching assumption”
Study pinched submanifolds in space forms, proving rigidity results.
problem Pinching condition on submanifolds in space forms.
method Analyzing geometry and topology under pinching conditions.
result Pinching condition forces homology to vanish or determines submanifolds up to congruence.
Sphere theorem extended without Ricci curvature positivity.
problem Eigenvalue pinching under Ricci curvature bounds.
method Generalization of Petersen and Aubry's sphere theorem.
result Eigenvalue pinching achieved without Ricci curvature positivity.
Sharp pinching conditions restrict the geometry and topology of submanifolds.
problem Understanding submanifolds under pinching conditions in arbitrary Riemannian manifolds.
method Analyzing submanifolds with pinching conditions involving second fundamental form and mean curvature.
result The pinching condition imposes strong geometric and topological restrictions on submanifolds.
Two spheres found with specific curvature constraints.
problem Existence of spheres with prescribed mean curvature.
method Proved existence of at least two embedded spheres with curvature satisfying pinching condition.
result Existence of at least two embedded spheres with prescribed mean curvature .
The study pinches the rigidity of self-shrinking surfaces in mean curvature flow.
problem Rigidity of self-shrinking hypersurfaces in mean curvature flow.
method Spectral upper-pinching theorem and weighted Poincaré estimate.
result Self-shrinking hypersurfaces are restricted to specific forms under certain conditions.
Classifies self-shrinkers in arbitrary dimensions under specific curvature conditions.
problem Classifying self-shrinkers with quadratic pinching conditions.
method Purely elliptic approach using weighted parabolicity, tailored to self-shrinkers.
result Generalized self-shrinking cylinders as solutions under quadratic pinching.
We study curvature pinching estimates of Ricci flow on complete 3- dimensional manifolds without bounded curvature assumption. We will derive some general curvature conditions which are preserved on any complete solution of 3-dim Ricci flow, these conditions include nonnegative Ricci curvature and sectional curvature a…
In this paper we prove that, under an explicit integral pinching assumption between the -norm of the Ricci curvature and the -norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein metric with positive curvature. In particular this implies that the manifold is diff…
Study sharp geometric and topological properties of pinched 4D submanifolds.
problem Pinched submanifolds in space forms.
method Four-dimensional geometry, Riemannian manifolds with nonnegative isotropic curvature, Bochner technique.
result Sharp results extend previous work without additional assumptions.
Estimates for hypersurfaces in CROSSes show cylindrical profiles near singularities.
problem Analyzing the mean curvature flow of hypersurfaces in complex or quaternionic projective spaces.
method Proving apriori estimates on principal curvatures under a pinching assumption.
result The asymptotic profile near a singularity is either strictly convex or cylindrical.
The paper studies neck-pinching of CP1-structures on surfaces, describing their limits.
problem Characterizing the degeneration of -structures on surfaces.
method Analyzing a path of -structures leaving every compact subset, converging holonomy in the PSL(2, C)-character variety.
result The limit of the path is described in terms of developing maps, holomorphic quadratic differentials, and pleated surfaces.
In this note we characterize compact hypersurfaces of dimension with constant mean curvature immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when and , they are locally contained in a rotational h…
We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the -norm of their scalar curvature and…
New findings on infinite subgroups in negatively curved spaces.
problem Characterizing infinite discrete isometry subgroups in negatively pinched Hadamard manifolds.
method Generalization of Bonahon's characterization to negatively pinched Hadamard manifolds.
result Every geometrically infinite isometry subgroup has a continuum of nonconical limit points.
Study a volume preserving flow using symmetric polynomials without curvature pinching assumptions.
problem Volume preserving flow of convex hypersurfaces without curvature pinching constraints.
method Power of the k-th elementary symmetric polynomial in principal curvatures.
result Solution exists for all times and converges to a round sphere in the volume preserving scalar curvature flow case.
The abstract discusses compactness of manifolds with pinched Ricci curvature.
problem Prove that a complete Riemannian manifold with positively pinched Ricci curvature is compact.
method Detailed alternate proof using quasi-conformal maps and mean curvature flow.
result Provides a proof of Hamilton's result on compactness of convex hypersurfaces.
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor…
Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.
problem Ergodicity and mixing properties of Anosov flows and their isometric extensions.
method Microlocal analysis of Pollicott-Ruelle resonances to study isometric extensions of Anosov flows.
result Ergodicity of frame flow on negatively-curved Riemannian manifolds under specific curvature assumptions.
We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the …
Study pinching constants for Kähler manifolds with positive curvature.
problem Pinching constants of Kähler manifolds with positive holomorphic sectional curvature.
method Apply techniques from Riemannian pinching theory to Kähler geometry.
result Prove a gap theorem for Kähler manifolds with almost quarter-pinched holomorphic sectional curvature.
Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.
problem Proving Hamilton's pinching conjecture for three-manifolds.
method Ricci flow with scale-invariant curvature decay and pinching preservation.
result Hamilton's pinching conjecture is proven without additional hypotheses.
Study pinched self-dual Weyl curvature in compact 4-manifolds.
problem Analyzing compact 4-manifolds with specific curvature properties.
method Examining harmonic self-dual Weyl curvature under pinching conditions.
result Characterized compact 4-manifolds with pinched self-dual Weyl curvature.
We refine a metric bunching estimate for pinched manifolds.
problem Improving an unstable bunching estimate for pinched metrics.
method Compact Riemannian manifolds with pointwise negatively pinched curvature tensor.
result Improved unstable bunching estimate.
Paper proves Hamilton's pinching theorem using mean curvature flow.
problem Hamilton's pinching theorem in extrinsic geometry.
method Mean curvature flow approach.
result Proof of Hamilton's pinching theorem.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
The paper proves conditions for four-dimensional gradient shrinking solitons to be flat or have specific curvature bounds.
problem Conditions for four-dimensional gradient shrinking solitons to be flat or have specific curvature bounds.
method Proving conditions for four-dimensional gradient shrinking solitons using Ricci curvature and Weyl curvature.
result Conditions for four-dimensional gradient shrinking solitons to be flat or have specific curvature bounds.
The paper pinches curvature in expanding Ricci solitons.
problem Curvature pinching in expanding Ricci solitons.
method Hamilton-Ivey type curvature pinching estimates.
result Three-dimensional Hamilton-Ivey type curvature pinching theorem.
This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.
problem Proving compactness for higher-dimensional manifolds with specific curvature conditions.
method Constructing Ricci flows for non-compact PIC1 pinched manifolds to prove compactness.
result Proves that PIC1 pinched manifolds of non-negative complex sectional curvature must be flat or compact.
Pinches flows in hyperbolic and de Sitter spaces with convex curvature.
problem Bounding dual flows in curved spaces.
method Proving pinching estimates with convex curvature function.
result Pinching estimates for dual flows in de Sitter space.
Compact Ricci solitons are shown to be round spheres under curvature pinching.
problem Characterizing compact gradient shrinking Ricci solitons under curvature pinching.
method Sharp algebraic curvature estimates, Yamabe-Sobolev inequality, and rigidity results.
result Compact gradient shrinking Ricci solitons are isometric to quotients of the round sphere under curvature pinching.
Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
Paper proves pinching theorem for minimal surfaces in spheres.
problem Pinching rigidity of minimal surfaces in spheres.
method Simon conjecture and Simons-type integral inequalities.
result New proof of pinching theorem for minimal surfaces in spheres.
Alternative proof of flatness for Ricci-pinched 3-manifolds.
problem Hamilton's pinching conjecture for 3-manifolds.
method Nonlinear potential theory with superquadratic volume growth.
result Flatness of Ricci-pinched 3-manifolds with superquadratic volume growth.
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.
Sphere bundles with 1/4-pinched metrics are induced by vector bundles.
problem Existence of 1/4-pinched metrics on sphere bundles.
method Proving all smooth sphere bundles with 1/4-pinched fiberwise metrics are induced bundles of vector bundles.
result Existence of many smooth n-sphere bundles without 1/4-pinched positively curved metrics.
Study conic 2-spheres, pinching curvature to find limits.
problem Curvature pinching of conic 2-spheres without Einstein metrics.
method Analyze metrics on conic 2-spheres, focusing on positive curvature and pinching constants.
result Best curvature pinching constant and explicit Gromov-Hausdorff limit when approached.
Rigidity theorem for pinched shrinking Ricci solitons.
problem Characterizing compact gradient shrinking Ricci solitons under pinching conditions.
method Proving isometricity to quotients of the round sphere using pinching conditions.
result Compact gradient shrinking Ricci solitons are rigid under $L^{ rac n2}$-pinching conditions.
Study shows pinched solutions of mean curvature flow blow up in codimension one.
problem Understanding blow-up behavior of pinched solutions in mean curvature flow.
method Analyzes blow-ups of compact solutions satisfying a pinching condition.
result Blow-ups of solutions must be codimension one.
The paper generalizes rigidity results for contact Anosov flows with bunching assumption.
problem Rigidity of contact Anosov flows in higher dimensions.
method Application of matching functions technique with bunching assumption.
result If two contact Anosov flows are conjugate, they are conjugate for some or even conjugate under additional assumptions.
Study submanifolds in spheres with Ricci curvature bounds.
problem Topology of submanifolds in spheres with Ricci curvature constraints.
method Investigates submanifolds in spheres with Ricci curvature lower bounds.
result Strong additional information on submanifold geometry.
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Study on G2-structures on solvmanifolds, focusing on Laplacian solitons and Ricci pinching.
problem Existence and interplay of Laplacian solitons and Ricci pinched G2-structures on solvmanifolds.
method Exploration of left-invariant G2-structures on solvable Lie groups, analysis of Ricci pinching properties.
result Obtained Ricci pinching properties and extremal values for G2-structures on solvmanifolds.
Ancient solutions to high codimension flow pinched by spheres.
problem Understanding ancient solutions to high codimension mean curvature flow.
method Showed compact ancient solutions with pinched second fundamental form must be shrinking spheres.
result Compact ancient solutions pinched by spheres are shrinking spheres.
Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
Study on surfaces pinched by curvature in space forms converging under specific conditions.
problem Investigating convergence of surfaces pinched by curvature in space forms.
method Proving convergence theorems for surfaces pinched by normal curvature in 4-dimensional space forms.
result Generalizes Baker-Nguyen's convergence theorem for surfaces pinched by curvature.
Study pinched submanifolds, proving homology vanishing results.
problem Understanding the geometry and topology of pinched submanifolds.
method Investigates submanifolds with a pinching condition on extrinsic invariants.
result Homology vanishing theorems for pinched submanifolds.