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.
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.
Sharp curvature pinching for mean curvature flow in spheres proved.
problem Proving sharp curvature pinching for mean curvature flow in spheres.
method Using blow-up arguments, codimension and cylindrical estimates, and rescaling.
result Smooth convergence to a totally geodesic limit in infinite time.
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 prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show …
Sharp estimates link curvature to topology, proving manifold rigidity.
problem Proving rigidity of manifolds under curvature pinching conditions.
method Sharp pointwise estimates and normalized Ricci flow.
result Proves manifold rigidity under strict sectional-scalar curvature pinching.
A sharp vanishing theorem for the Lp cohomology torsion of Riemannian manifolds with pinched negative curvature is given. It follows that certain negatively curved homogeneous spaces cannot be quasiisometric to better pinched manifolds.
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.
For a convex domain D bounded by the hypersurface ∂D in a space of constant curvature we give sharp bounds on the width R−r of a spherical shell with radii R and r that can enclose ∂D, provided that normal curvatures of ∂D are pinched by two positive constants. Furthermore, in the …
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…
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.
Develops local curvature estimates for mean curvature flow.
problem Sharp curvature pinching estimates for mean curvature flow.
method Local version of Huisken-Stampacchia iteration.
result Local curvature estimates do not depend on noncollapsing quality.
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.
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.
We give a topological interpretation of the space of L2-harmonic forms on finite-volume manifolds with sufficiently pinched negative curvature. We give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants. The me…
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…
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.
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.
In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere…
Study on 4D solitons with curvature constraints.
problem Characterizing gradient shrinking Ricci solitons with positive modified sectional curvature.
method Sharp pinching conditions, weighted integral gap results, Hitchin-Thorpe inequality.
result Locally Kähler property under specific curvature conditions.
We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature ('cylindrical estimates')…
In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the epsilon-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C.
In this paper, we will prove a gap theorem for four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying a pinched Weyl curvature, either is flat, or λ1+λ2≥c0R>0 everywhere f…
Let Mn(n≥3) be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that Mn satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4,…
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. 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. 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 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.
Let Fn+p(c) be an (n+p)-dimensional simply connected space form with nonnegative constant curvature c. We prove that if Mn(n≥4) is a compact submanifold in Fn+p(c), and if RicM>(n−2)(c+H2), where H is the mean curvature of M, then M is homeomorphic to a sphere. We also show that the pinchi…
We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces i…
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 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.
Estimates spectral projections restricted to uniformly embedded submanifolds.
problem Estimating spectral projections on submanifolds of manifolds with nonpositive curvature.
method Estimates the L2(M)oLq(Σ) norm of spectral projection operators. result Sharp spectral projection estimates for small spectral windows.
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.
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…
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.
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 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.
For a Riemannian metric g on the two-sphere, let ℓmin(g) be the length of the shortest closed geodesic and ℓmax(g) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g…