The paper studies knot complements and their representations with special attention to pinched octahedra.
problem Understanding knot complements and their representations with pinched octahedra.
method Analyzing pseudo-hyperbolic structures and w-variables in knot complements. result A w-solution with pinched octahedra induces a new knot representation. This study finds the shape of centrally symmetric octahedra with specific angles.
problem Finding the shape of centrally symmetric octahedra with prescribed cone-deficits.
method Inspired by Thurston's work, the paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits, showing it forms a real hyperbolic ideal tetrahedron.
result The set of shapes of centrally symmetric octahedra with prescribed cone-deficits forms a real hyperbolic ideal tetrahedron with dihedral angles half of the prescribed cone-deficits.
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary …
The paper studies pseudo-hyperbolic structures derived from knot complements.
problem Understanding geometric and combinatorial properties of knot complements.
method Using ideal octahedral decompositions and pseudo-hyperbolic structures, the paper computes complex volumes and cusp shapes of knots and links.
result Concrete formulas for Wirtinger generators and cusp shapes are derived, and explicit solutions are provided for various knots.
Geometric proof confirms link volume conjecture.
problem Volume conjecture for hyperbolic 3-manifolds.
method Topological and skein theoretic techniques.
result Complements of octahedral fully augmented links are isometric to fundamental shadow links.
We prove that every complete finite-volume hyperbolic 3-manifold M that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold W, which is also tessellated into right-angled regular pol…
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.
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.
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…
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.
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.
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.
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.
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.
Study pinches intrinsic and normal curvatures of minimal surfaces in a sphere.
problem Pinching constraints on intrinsic and normal curvatures of minimal surfaces.
method Established orthonormal frame field, derived property K+KN=1, used to pinch curvatures. result Pinched constraints on intrinsic and normal curvatures of minimal surfaces.
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. Explains the Borromean rings, icosahedron, and Poincaré homology sphere.
problem Exploring the relationship between Borromean rings, icosahedron, and Poincaré homology sphere.
method Introduction of topological concepts and geometric construction of icosahedral compound of octahedra.
result Proofs about the orientation-preserving symmetry group of an icosahedron and the linked nature of Borromean rings.
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.
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 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.
Study pinches gradient solitons using curvature estimates.
problem Integral pinching rigidity of gradient shrinking solitons.
method Algebraic curvature estimates and Yamabe-Sobolev inequality.
result Proves integral pinching rigidity for compact gradient shrinking solitons.
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.
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.