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.
The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial a…
The famous pinching problem says that on a compact simply connected n-manifold if its sectional curvature satisfies Kmin>(1/4)Kmax>0, then the manifold is homeomorphic to the sphere. In [8, problem 12], S. T. Yau proposed the following problem: If we replace Kmax by the scalar curvature, can we deduc…
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.
The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.
problem Improving inequalities for Kähler-Einstein manifolds.
method Using invariant theory and curvature conditions to express and improve inequalities.
result Improved inequalities for Kähler-Einstein manifolds with smaller pinching constants.
The paper establishes eigenvalue inequalities for a specific operator on curved spaces.
problem Eigenvalue estimation for a specific operator on curved domains.
method Bochner type formula and Rauch comparison theorem.
result Universal inequalities for eigenvalues of the drifted Cheng-Yau operator.
Study shows how neck pinches occur in Lagrangian flows and their continuation.
problem Understanding and continuation of Lagrangian mean curvature flows with singularities.
method Analyzes zero Maslov, rational Lagrangian flows in compact Calabi-Yau surfaces.
result Tangent flow is unique and can be continued past singularities.
The study proves the non-existence of certain Kähler metrics with specific curvature properties.
problem Non-existence of complete Kähler metrics with negatively pinched holomorphic sectional curvature.
method Construction of a Kähler metric with negatively pinched holomorphic sectional curvature and application of equivalence of invariant metrics.
result The dichotomy of completeness and non-existence of Kähler metrics with negatively pinched holomorphic sectional curvature.
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.
Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mn is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R0>σnKmax, where σn∈(41,1) is an explicit positive constan…
For a manifold with nonpositive curvature, the Martin boundary is described by the behavior of normalized Green's functions at infinity. A classical result by Anderson and Schoen states that if the manifold has pinched negative curvature, the geometric boundary is the same as the Martin boundary. In this paper, we stud…
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.
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. Among closed G2-structures there are two very distinguished classes: Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the existence problem and explore possible interplays between these concepts in the context of left-invariant G2-structures on solvable Lie groups. Also, some Ricci pinching prope…
Estimates gaps between eigenvalues for elliptic operators on manifolds.
problem Estimating the gaps between consecutive eigenvalues for elliptic differential operators.
method Analyzes a class of second-order elliptic differential operators in divergence form with Dirichlet boundary conditions.
result Estimates for the upper bound of gaps between eigenvalues, with results matching known best estimates for specific cases.
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…
Solves Plateau problem for surfaces in pinched curvature manifolds.
problem Asymptotic Plateau problem for immersed surfaces in pinched curvature manifolds.
method Complete solution to asymptotic Plateau problem, providing dynamical stability of hypersurface laminations.
result Achieved complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds.
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.
Volume gaps for minimal submanifolds in spheres are proven.
problem Volume gaps for minimal submanifolds in spheres.
method Analyzing height functions and multiplicity in direction p.
result Volume inequalities for minimal submanifolds in spheres.
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.
We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume g…
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.
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.
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.
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.
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.
Researchers solved the even Lp-Minkowski problem under curvature pinching.
problem Solving the even Lp-Minkowski problem under curvature pinching. method Anisotropic Riemannian metric comparison and anisotropic curvature analysis.
result The even Lp-Minkowski inequality and uniqueness are proven for all p≥pγ. 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.
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.
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. Lower bound for Steklov eigenvalues on negatively curved manifolds.
problem Finding a geometric lower bound for the first nonzero Steklov eigenvalue.
method Combining a uniform lower bound for the first eigenvalue of the Steklov-Dirichlet problem and a tubular neighborhood theorem for totally geodesic hypersurfaces.
result A geometric lower bound for the first nonzero Steklov eigenvalue in terms of total and boundary volumes.
The paper solves the Nirenberg problem on high-dimensional half spheres with pinching conditions.
problem Finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on half spheres.
method Variational approach with pseudogradient and Morse theory to handle non-compactness.
result Existence results for the Nirenberg problem under various pinching 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 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.