Study integral bounds for submanifolds in low codimension with topological implications.
problem Integral curvature bounds and topological obstructions for submanifolds.
method Integral curvature bounds in terms of Betti numbers, δ-pinched immersions, and pinched second fundamental form. result Obtained topological obstructions for δ-pinched immersions and intrinsic obstructions for minimal submanifolds in spheres. Proves a pinching theorem for self-shrinkers of mean curvature flow.
problem Pinch on the squared norm of the second fundamental form of self-shrinkers.
method Proves a theorem for n−dimensional closed self-shrinkers. result Closed self-shrinkers must be the standard sphere if pinched.
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.
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.
Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.
problem Establishing a lower bound for the squared length of the second fundamental form on minimal hypersurfaces.
method Analyzes closed embedded, non-totally geodesic minimal hypersurfaces in Sn+1, proving a positive constant δ(n) depending only on n. result Introduces a positive constant δ(n) such that ∫MS≥δ(n)mVol(Mn) for any minimal hypersurface Mn in Sn+1. 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.
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.
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 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.
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.
Sharp estimate for genus of embedded surfaces in 3-sphere.
problem Estimating the genus of embedded surfaces in the 3-sphere.
method Refined volume estimate and pinching method on the norm of traceless second fundamental form.
result Sharp pinching estimate for the genus of a surface in S3. 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.
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.
We prove an extension of a theorem of A.Ros on a characterization of seven compact Kaehler submanifolds by holomorphic pinching to certain submanifolds of the complex Grassmannian manifolds.
The paper proves new pinching theorems for self-shrinkers and λ-hypersurfaces.
problem Characterizing complete self-shrinkers and λ-hypersurfaces with specific curvature conditions.
method Verification of pinching theorems for self-shrinkers and λ-hypersurfaces with polynomial volume growth.
result Conditions under which the curvature of self-shrinkers and λ-hypersurfaces are pinched to specific values.
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.
The Hopf fibration is rigid among minimal maps between spheres.
problem Characterizing minimal submersions between spheres.
method Analyzing the properties of the Hopf fibration and minimal maps.
result The Hopf fibration is the only minimal submersion from S3 to S2 under certain conditions. Sharp curvature estimates for mean curvature flow in spheres.
problem Understanding the behavior of surfaces evolving under mean curvature flow in spheres.
method Proving asymptotically sharp curvature pinching estimates and using them to derive derivative and convexity estimates.
result Partial classification of singularity models and new rigidity results for ancient solutions.
We show that, among free boundary minimal surfaces in the unit ball in the three-dimensional Euclidean space, the flat equatorial disk and the critical catenoid are characterised by a pinching condition on the length of their second fundamental form.
We investigate the evolution of closed strictly convex hypersurfaces in Rn+1, n=3, for contracting normal velocities, including powers of the mean curvature, of the norm of the second fundamental form, and of the Gauss curvature. We prove convergence to a round point for 2-pinched initial hypersurfaces. I…
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.
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 some pinching results for the extrinsic radius of compact hypersurfaces in space forms. We show that if the pinching condion is strong enough with a dependance on the norm of the second foundamental form, then the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.
In this paper, we give a Simons' type formula for the cmc surfaces in homogeneous 3-manifolds E(κ,τ), τ=0. As an application, we give a rigidity result in the case of κ>4τ2 for the cmc surfaces under a pinching assumption of the second fundamental form.
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.
We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a …
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.
The paper studies essential spectra of submanifolds in Euclidean spaces.
problem Investigating the essential spectrum of submanifolds under geometric conditions.
method Analyzing submanifolds in Euclidean spaces with various geometric constraints.
result The essential spectrum of a complete non-compact submanifold is [0,+∞) if the second fundamental form satisfies certain Lp norms. Sharp pinching theorem for submanifolds in spheres.
problem Characterizing submanifolds in spheres based on curvature bounds.
method Conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.
result Complete submanifolds with specific curvature bounds are either totally geodesic or Clifford tori/Veronese surfaces.
Extends curvature gap characterization for minimal surfaces in a ball.
problem Characterization of minimal surfaces in a ball.
method Pinching condition on second fundamental form.
result Extension to higher codimension.
For a compact minimal hypersurface M in Sn+1 with the squared length of the second fundamental form S we confirm that there exists a positive constant $\de(n)$ depending only on n, such that if n≤S≤n+δ(n), then S≡n, i.e., M is a Clifford minimal hypersurface, in particular, when $n\ge 6,…
In this paper, we give pinching Theorems for the first nonzero eigenvalue λ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is 1 then, for any ε>0, there exists a constant C_ε depending on the dimension n of M and the L_∞-norm of the …
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.
In this paper, we study the evolution of submannifold moving by mean curvature minus a external force field. We prove that the flow has a long-time smooth solution for all time under almost optimal conditions. Those conditions are that the second fundamental form on the initial submanifolds is not too large, the extern…
We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Gold…
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
The paper studies hypersurfaces in 5D space forms with topological and rigidity results.
problem Characterizing and bounding hypersurfaces in 5D space forms.
method Analyzing the Weyl tensor, deriving topological bounds, and using integral inequalities.
result Sharp topological bounds on the Weyl functional for closed, minimal hypersurfaces.
Flat minimal tori counterexamples refute Lu's second-gap conjecture.
problem Lu's second-gap conjecture about minimal surfaces in higher codimensions.
method Constructing closed embedded counterexamples for minimal surfaces.
result Constant values of S+λ2 realized by flat minimal tori are dense in (2,3), refuting the conjecture. We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let Mn be a compact hypersurface with constant mean curvature H in Sn+1. Denote by S the squared norm of th…
The paper explores non-minimal solitons in the sphere with unique properties.
problem Exploring soliton solutions of mean curvature flow in the unit sphere.
method Analyzing integral curves of the Hopf vector field and using symmetry properties.
result A non-minimal, complete example with topology S2n−1imesR. The study pinches rigidity theorems for minimal submanifolds in spheres.
problem Pinching rigidity theorems for minimal submanifolds in spheres.
method Analyzes the shape operators and eigenvalues of submanifolds to prove rigidity conditions.
result If certain conditions are met, the normal bundle of the submanifold is flat.
Let M be a closed 5-manifold of pinched curvature 0<δ\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) δ=1/4 and the fundamental group is a non-cyclic group of order at least C, a constant. (ii) The center of the fundamental group has in…
We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the Möbius band, and the products of a line with the total spaces of flat vector bundles over closed infranilmanifolds.
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 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.
The paper studies the blow-up of conformal mean curvature flow in higher codimension.
problem Analyzing the blow-up behavior of conformal mean curvature flow.
method Introduced and studied conformal mean curvature flow, derived blow-up theorem and evolution formulas.
result Maximum of the square norm of the second fundamental form tends to infinity in finite time.
In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form A under specific conditions. result The normalized second fundamental form A is intrinsic if σ2k+1(A)eq0 for some k≥1.