Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
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The paper proves sphere theorems for submanifolds in Kähler manifolds.
problem Sphere theorems for submanifolds in Kähler manifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler manifolds, especially in complex space forms.
result Proves sphere theorems for submanifolds in Kähler manifolds.
Sphere theorems for submanifolds in Kähler and Sasaki spaces.
problem Proving sphere theorems for Lagrangian and Legendrian submanifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler and Sasaki spaces.
result Proved sphere theorems for Lagrangian and Legendrian submanifolds.
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
Proves a theorem connecting graph theory spheres, reformulating Morse conditions.
problem Defines and connects spheres in graph theory.
method Proves a theorem bridging two graph theory sphere definitions.
result Reformulates Morse conditions using center manifolds and level surface graphs.
The paper establishes a new sphere theorem for certain types of manifolds.
problem Finding conditions under which compact manifolds are spheres.
method Developed a generalized sphere theorem for manifolds with radial Ricci curvature.
result Established conditions for compact manifolds to be topologically spheres.
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
Sphere theorems for specific manifolds with curvature constraints.
problem Sphere theorems for Riemannian manifolds with scalar curvature bounds and non-collapsed spaces.
method Analysis of scalar curvature and mean distance constraints.
result Established sphere theorems for the specified manifolds.
Sphere theorem extended without Ricci curvature positivity.
problem Eigenvalue pinching under Ricci curvature bounds.
method Generalization of Petersen and Aubry's sphere theorem.
result Eigenvalue pinching achieved without Ricci curvature positivity.
Simplified proof of Lefschetz theorem for PL spheres.
problem Proving the Lefschetz theorem for PL spheres.
method Using Pachner's Theorem to replace geometric constructions.
result Simpler proof and implications for g-conjecture and Kalai-Sarkaria conjecture.
The study proves the existence of geodesics on reversible Finsler spheres.
problem Existence of closed geodesics on Finsler 2-spheres.
method Generalization of Grayson's curve shortening flow.
result Existence of three simple closed geodesics and infinitely many closed geodesics.
Mean curvature flow of arbitrary codimension in spheres and sharp differentiable sphere theoremmath.DG
In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [30] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [2] as well as the differentiable sphere theorems of Gu-Xu-Zhao [16, 50, 52].
New theorem on spheres with punctures using infinity metric.
problem Rigidity of metrics on spheres with punctures.
method Proof of Llarull's theorem for metrics on spheres with finitely many points removed.
result The rigidity theorem holds for metrics on spheres with finitely many points removed.
New theorem for doodles on sphere, similar to Markov's.
problem Understanding doodles on a sphere.
method Description of twins with equivalent closures.
result Analogous to Markov's theorem for doodles.
New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theorymath.DG
Survey on mean curvature flow with sphere theorems and Yau rigidity theory.
problem Sphere theorems for submanifolds with arbitrary codimension.
method Recent developments on convergence theorems for mean curvature flow.
result Optimal convergence theorem for arbitrary codimension mean curvature flow.
Generalizes Hopf degree theorem to nontrivial bundles.
problem Classifying maps from manifolds to spheres.
method Generalization of Hopf degree theorem to nontrivial bundles.
result Classifies sections of nontrivial -sphere bundles.
Green functions for GJMS operators on spheres derived, linking geometry and rigidity.
problem Deriving Green functions for GJMS operators on spheres.
method Explicit representation formulae derived using Gegenbauer polynomials.
result Spheres uniquely characterized by their Green functions, with strong rigidity theorems for .
The paper proves a maximal diameter sphere theorem for certain Riemannian manifolds.
problem Bounding the diameter of Riemannian manifolds with radial sectional curvature.
method Proving a theorem for a wide class of two-sphere of revolution manifolds.
result The diameter of a manifold equals that of a sphere if and only if the manifold is isometric to the sphere.
Sphere Theorems in Geometrymath.DG
In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton'…
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
Smooth approximations of Lipschitz maps via Ehresmann fibrations and Reeb sphere theorem for functions.
problem Approximating Lipschitz maps and understanding singular points in Riemannian manifolds.
method Using Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions.
result A Lipschitz map can be approximated by a smooth map via Ehresmann fibrations.
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.
Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.
problem Existence and non-existence of specific geodesic nets on flat spheres.
method The theorem of Gauss-Bonnet is applied to demonstrate results.
result Existence and non-existence of geodesic nets on regular doubled polygons.
New -hypersurfaces not isometric to standard spheres.
problem No Alexandrov theorem for -hypersurfaces.
method Constructing compact embedded -hypersurfaces diffeomorphic to a sphere.
result Found -hypersurfaces not isometric to standard spheres.
Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
Computer proof verifies key differential properties in sphere classifications.
problem Verifying holomorphic properties of meromorphic differentials in sphere classifications.
method Computer-assisted proof using Sage software.
result Holomorphicity of quartic and octic differentials confirmed.
The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.
problem Which min-max widths of the unit 3-sphere lie between and ?
method Homological min-max theory and stronger versions of multiplicity one theorems.
result Proves the 10th to 13th min-max widths of the unit 3-sphere lie between and .
Characterizes simplicial complexes embedding into spheres with few vertices.
problem Characterizing simplicial complexes that embed into spheres with few vertices.
method Simple characterization using non-face families and analogy with Fáry's theorem.
result Recovery of van Kampen--Flores theorem and Erd\H os--Ko--Rado theorem.
The paper examines rigidity of special submanifolds in spheres with curvature constraints.
problem Rigidity of -extremal submanifolds in a sphere under curvature conditions.
method Proves pinching theorems for submanifolds with various curvature conditions.
result Various curvature conditions lead to rigidity of -extremal submanifolds.
Sphere theorems proved for manifolds with specific curvature conditions.
problem Sphere theorems for Riemannian manifolds with curvature operator constraints.
method Investigation of eigenvalues and curvature operator conditions.
result Proved sphere theorems in dimensions three and four, homological sphere theorem in higher dimensions.
New theorem shows nearly spherical manifolds can be mapped from spheres.
problem Generalizing Caffarelli's theorem to nearly spherical manifolds.
method Optimal transport map on the sphere, stability result.
result Every nearly spherical manifold can be mapped from a sphere.
Simplified proof of Cerf's theorem on 3-sphere diffeomorphisms.
problem Proving the connectedness of direct diffeomorphisms of the 3-sphere.
method Rigidity property of foliations defined by non-vanishing closed one-forms.
result Connected group of direct diffeomorphisms of the 3-sphere.
The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.
problem Proving sphere theorems for hypersurfaces with regularity.
method Extending Montiel-Ros argument and using Legendrian cycles.
result Proves existence of -dimensional Legendrian cycles with -dimensional support.
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.
The paper proves a gap theorem for special harmonic maps between spheres.
problem Understanding the behavior of -harmonic maps between spheres.
method Analysis of approximations and energy identities to derive a gap theorem.
result An optimal gap theorem for -harmonic maps of specific degrees.
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.
From radial curvature geometry's standpoint, we prove a sphere theorem of the Grove-Shiohama type for a certain class of compact Finsler manifolds.
In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in .
The paper constructs metrics on spheres with families of minimal hypersurfaces.
problem Finding Riemannian metrics on spheres with specific families of minimal hypersurfaces.
method Used Nash-Moser Inverse Function Theorem in the tame maps setting.
result Generalized Guillemin's theorem for Zoll families of minimal hypersurfaces.
Sphere theorems for p-Laplacian eigenvalues established.
problem Sphere theorems for p-Laplacian eigenvalues.
method Established sphere theorems for p-Laplacian eigenvalues.
result Sphere theorems for p-Laplacian eigenvalues established.
Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
Softens classical theorems on Ricci curvature.
problem Classical theorems on Ricci curvature with strict hypotheses.
method Soft, quantitatively optimal extensions in -topology.
result Optimal extensions to Ricci curvature theorems.
Study eigenvalues of ellipsoids near a sphere, comparing to sphere's.
problem Analyzing changes in Laplacian eigenvalues for ellipsoids near a sphere.
method Comparison with standard Euclidean unit sphere, under Gaussian curvature condition.
result Eigenvalues of ellipsoids near a sphere, with comparison to sphere's.
Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two open problems for closed 3 manifolds with positive Ricci curvature since they are b…
3D spheres with certain properties approach the round sphere.
problem Flexibility of Llarull's Theorem in dimension 3.
method Proof based on spacetime harmonic functions.
result 3D spheres with bounded Cheeger isoperimetric constant and scalar curvatures tending to 6 approach the round sphere.
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
Proves a concordance version of a 4D light bulb theorem.
problem Proving concordance of embedded spheres in 4-manifolds.
method Analyzes homotopically embedded 2-spheres in 4-manifolds with specific conditions.
result Concordance of spheres under given conditions.
Extends Borsuk-Ulam theorem with applications in sphere coverings and colorings.
problem Complexity bounds and structural insights for triangulated sphere mappings.
method Combinatorial labeling and order type analysis of finite point sets.
result New topological Hall theorem and generalizations of hypergraph Hall theorems.