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.
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.
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.
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.
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 pinches conditions for minimal surfaces in hyperbolic space and hemisphere.
problem Characterizing minimal surfaces with free boundary in hyperbolic space and hemisphere.
method Pinching condition involving second fundamental form, support function, and potential function.
result Characterization of totally geodesic disk and rotational annulus.
The paper examines ancient solutions of mean curvature flow in space forms with curvature pinching conditions.
problem Investigating rigidity of ancient solutions of mean curvature flow in space forms.
method Sharp asymptotic pointwise curvature pinching conditions and asymptotic integral curvature pinching conditions.
result Ancient solutions in a sphere are either a shrinking spherical cap or a totally geodesic sphere, and in a hyperbolic space, they are a family of shrinking spheres.
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.
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.
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.
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 prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous convergence theorem due to Huisken [9]. Moreover, we prove a convergence theorem und…
Study pinched submanifolds in symmetric spaces, proving flow behaviors.
problem Analyzing mean curvature flow in symmetric spaces.
method Proved flow behaviors for pinched submanifolds in rank one symmetric spaces.
result Submanifolds in symmetric spaces either collapse or converge smoothly.
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.
New G2-structures found on Lie groups with strong structural conditions.
problem Existence and structure of extremally Ricci pinched G2-structures on Lie groups.
method Strong structural conditions on Lie algebra, deformation and rigidity studies.
result Three new examples of extremally Ricci pinched G2-structures, all steady Laplacian solitons.
The paper studies how submanifolds of a sphere evolve over time.
problem Evolution of pinched submanifolds in the sphere.
method High codimension mean curvature flow with pinching conditions.
result Convergence to a round point or totally geodesic sphere under pinching conditions.
The study pinches conditions for constant mean curvature surfaces in convex 3-manifolds.
problem Understanding the topology and geometry of constant mean curvature surfaces with free boundaries in convex 3-manifolds.
method Provided pinching conditions on the traceless second fundamental form to guarantee surface topology.
result The surface is either a disk, annulus, spherical cap, or Delaunay surface under certain conditions.
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.
New proof confirms noncompact locally conformally flat manifolds are compact.
problem Rigidity of Schouten tensor under conformal transformations.
method Proof of Cheng's theorem using modified Schouten tensor.
result Noncompact locally conformally flat manifolds are compact.
We consider the evolution by mean curvature flow of Lagrangian submanifolds of the complex projective space CP^n. We prove that, if the initial value satisfies a suitable pinching condition, then the flow exists for all times and the manifold converges to a totally geodesic submanifold. As a corollary, we obtain that a…
New existence results for curvature problem on balls with specific conditions.
problem Existence of solutions for a prescribed mean curvature problem on a ball.
method Combining critical points at infinity approach with Morse theory.
result New existence results for higher dimensional case n≥5 under pinching conditions. Study new Ricci flow invariant curvature conditions.
problem Topology of manifolds with pinched curvature.
method Provide quantitative evidence for an unpublished conjecture.
result Topology of manifolds with pinched curvature studied.
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 h satisfying pinching condition. result Existence of at least two embedded spheres with prescribed mean curvature h. A 3-manifold's Ricci pinching condition implies it's flat if it has Euclidean volume growth.
problem Understanding the flatness of 3-manifolds under Ricci pinching conditions.
method Alternative proof using potential theory.
result If a 3-manifold has Euclidean volume growth and satisfies the Ricci pinching condition, it is flat.
The paper proves conditions for a manifold to be homeomorphic to a spherical space form.
problem Proving conditions for a manifold to be homeomorphic to a spherical space form.
method Proving conditions using curvature inequalities for orthonormal four-frames.
result The manifold is homeomorphic to a spherical space form under the given curvature condition.
Paper finds critical metrics with pinched curvature are geodesic balls.
problem Identifying critical metrics with specific curvature constraints.
method Proved isometry to geodesic balls in S^n and provided conditions for the gradient of the potential function.
result Critical metrics with pinched curvature are isometric to geodesic balls in S^n.
Ancient geometric flows of submanifolds are characterized under curvature pinching.
problem Characterizing ancient solutions of geometric flows under curvature constraints.
method Rigidity theorems for ancient solutions of geometric flows of immersed submanifolds.
result Pinching conditions on the second fundamental form characterize the shrinking sphere for mean curvature flow in higher codimensions and certain nonlinear curvature flows of hypersurfaces.
We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.
We prove that a n-dimensional, 4≤n≤6, compact gradient shrinking Ricci soliton satisfying a Ln/2-pinching condition is isometric to a quotient of the round Sn. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result f…
In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.
We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost …
New curvature condition proves rigidity of Bryant Ricci solitons.
problem Proving rigidity of Bryant Ricci solitons.
method Introducing a new curvature-pinching condition and proving rigidity results.
result Rotationally symmetric solutions of steady Ricci solitons are rigid under the new curvature condition.
Compact Bach-flat manifolds with positive σ2 are Einstein if curvature pinches.
problem Characterizing compact Bach-flat manifolds with positive σ2. method Proving compact Bach-flat manifolds with positive σ2 are Einstein under curvature pinching conditions. result Compact Bach-flat manifolds with positive σ2 are Einstein if curvature pinches. The study examines how high powers of curvature affect the expansion of pinched hypersurfaces in Euclidean and hyperbolic spaces.
problem The preservation and convergence of pinched hypersurfaces under high powers of curvature flows.
method Proving convergence results for expanding curvature flows with flow speeds of the form F−p, where p>1 and F is a curvature function. result A pinching condition is preserved, and properly rescaled hypersurfaces converge to the unit sphere.
In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher order mean curvatures. We show that under a suitable pinching condition, the hyp…
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…
Study mean curvature flow to prove submanifolds of spheres are diffeomorphic.
problem Prove submanifolds of spheres are diffeomorphic under curvature pinching conditions.
method Use mean curvature flow with surgeries to prove diffeomorphism.
result Prove any smoothly, properly immersed submanifold of SKn+1 satisfying the pinching condition is diffeomorphic to Sn or connected sum of handles. Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.
problem Rigidity of minimal Legendrian submanifolds in unit Euclidean spheres.
method Using Lu's inequality and eigenvalues of fundamental matrices to establish pinching theorems.
result Optimal pinching theorem and rigidity theorem for submanifolds of all dimensions.
We prove pinching estimates for solutions of the linearized Ricci flow system on a closed manifold of dimension n≥4 with positive scalar curvature and vanishing Weyl tensor. If the vanishing Weyl tensor condition is removed, we only give a rough pinching estimate controlled by some blow-up function in a short tim…
Paper proves cohomology vanishing theorems for submanifolds under certain conditions.
problem Establishing cohomology vanishing theorems for submanifolds with specific geometric constraints.
method Using a new Hardy type inequality, the authors prove vanishing theorems for submanifolds with pinching conditions.
result The paper removes the condition on the flatness of the normal bundle and partially answers questions on optimal pinching constants.
Study approximates product of spheres using Laplacian eigenvalues.
problem Approximating product of spheres using Laplacian eigenvalues.
method Gromov-Hausdorff approximation with pinching condition on eigenvalues.
result Convergence to product of spheres achieved.
We prove that an n-dimensional, n≥4, compact gradient shrinking Ricci soliton satisfying a L2n-pinching condition is isometric to a quotient of the round Sn, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).
Estimates eigenvalue for manifolds with specific forms under certain conditions.
problem Estimating the first eigenvalue of Laplacian on manifolds with almost parallel p-forms. method Uses Lichnerowicz-Obata estimate and pinching conditions to analyze eigenvalues.
result Establishes a Lichnerowicz-Obata type estimate for the first eigenvalue.
Classification of G2-structures on Lie groups with Ricci pinched conditions.
problem Classifying G2-structures on Lie groups under specific geometric conditions.
method Complete classification of left-invariant closed G2-structures on Lie groups, extremally Ricci pinched, up to equivalence and scaling.
result Five distinct G2-structures on five different completely solvable Lie groups, with one unimodular case being exact.
The paper studies hypersurfaces in spheres using mean curvature flow with surgery.
problem Studying hypersurfaces in spheres under specific curvature pinching conditions.
method Using mean curvature flow with surgery to preserve and analyze curvature pinching conditions.
result Hypersurfaces satisfying the pinching condition are diffeomorphic to spheres or connected sums of spheres.
In this paper, we give the full proof of a conjecture of R.Hamilton that for (M3,g) being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where R>0 is the positive scalar curvature and $\ep>0$ is a uniform constant, M3 is compact. One of the key i…
We consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K of M so that the outward normal exponential map off of the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactif…
The paper examines rigidity of special submanifolds in spheres with curvature constraints.
problem Rigidity of k-extremal submanifolds in a sphere under curvature conditions. method Proves pinching theorems for submanifolds with various curvature conditions.
result Various curvature conditions lead to rigidity of k-extremal submanifolds.