Study pinching sequences to understand degeneration of anti-de Sitter structures.
problem Understanding the degeneration of anti-de Sitter structures along pinching sequences.
method Parameterization of deformation space and analysis of pinching sequences.
result Regular anti-de Sitter structures appear as limiting points.
We show that in each dimension n≥10 there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian n-manifolds with 0≤sec≤1, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type whic…
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.
Sphere theorems extended to Riemannian foliations with new results on curvature and leaf spaces.
problem Sphere theorems for Riemannian foliations with transverse curvature constraints.
method Deformation theory and Gromov-Hausdorff limits to prove sphere theorems.
result Complete Riemannian foliations with quarter-pinched transverse sectional curvature develop to simple foliations.
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 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.
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.
In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space Rn+1 with positive mean curvature is κ-noncollapsing, and a blow-up sequence conve…
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.
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.
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…
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 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 connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.
problem Understanding and comparing different types of Lagrangian fillings of Legendrian weaves.
method Establishing new Reidemeister moves and combinatorial isotopies between Lagrangian fillings, comparing sheaf quantizations.
result Legendrian weaves generalize previously known methods to produce infinitely many distinct Lagrangian fillings.
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.
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.
The study finds optimal curvature pinching in Heintze groups.
problem Exploring curvature properties in Heintze groups.
method Examining metric properties of rank-one symmetric spaces, proving existence of metrics on Heintze groups of Carnot-type.
result Optimal curvature pinching is demonstrated in a special case.
Ricci flow on flat manifolds converges to Euclidean space under curvature pinching.
problem Curvature pinching on asymptotically flat manifolds.
method Ricci flow on asymptotically flat manifolds with integral curvature pinching.
result Ricci flow converges to flat Euclidean space for sufficiently pinched curvature.
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.
Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.
problem Understanding Weyl curvature pinching on 4-manifolds.
method Analyzing harmonic and pinched self-dual Weyl curvature, proving anti-self-duality.
result Proves anti-self-duality for compact 4-manifolds with pinched self-dual Weyl curvature.
New restrictions found on 4-manifolds with pinched curvature.
problem Restrictions on Euler characteristic and signature of 4-manifolds with pinched curvature.
method Proved new restrictions on Euler characteristic and signature of oriented 4-manifolds with pinched sectional curvature.
result Simply connected 4-manifolds with δ≤sec≤1 are homeomorphic to S4 or CP2. In this paper, the pinching problems of complete λ-hypersurfaces in a Euclidean space Rn+1 are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete λ-hypersurfaces in a Euclidean space Rn+1.
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.
Study pinches volume of CAT(1) spaces, proving sphere theorem and manifold recognition criterion.
problem Volume pinching problems in CAT(1) spaces.
method Characterization of compact geodesically complete CAT(1) spaces, sphere theorem proof, manifold recognition criterion formulation.
result Sphere theorem for compact CAT(1) homology manifolds of small volume, manifold recognition criterion under upper curvature bound.
The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.
problem Quantifying rigidity in manifolds with nonnegative Ricci curvature.
method Investigates pinching of Colding's monotone functionals and constructs k-splitting functions. result Quantitative control of splitting functions by pinching at independent points controls the distance to the nearest cone.
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.
The paper constructs a new metric on Kähler manifolds.
problem Finding metrics with specific curvature properties on Kähler manifolds.
method Constructing an almost negatively 1/4-pinched Riemannian metric.
result First known examples of not locally symmetric Kähler manifolds with this metric.
Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.
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.
Proves Hamilton's theorem using mean curvature flow.
problem Compactness of pinched hypersurfaces with bounded curvature.
method Mean curvature flow to prove Hamilton's theorem.
result Rigorous proof of Hamilton's theorem.
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.
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.
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 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 …
In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.
Given a closed contact 3-manifold with a compatible Riemannian metric, we show that if the sectional curvature is 1/4-pinched, then the contact structure is universally tight. This result improves the Contact Sphere Theorem in [EKM12], where a 4/9-pinching constant was imposed. Some tightness results on positively curv…
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.
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.