arXiv research

Sharp convergence theorem for sphere submanifolds proved.

The paper establishes a new sphere theorem for certain types of manifolds.

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

Sphere theorems for specific manifolds with curvature constraints.

In this paper, we prove some differentiable sphere theorems and topological sphere theorems for submanifolds in Kähler manifold, especially in complex space forms.

In this paper, we prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in Kähler manifold and Legendrian submanifolds in Sasaki space form.

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.

Generalizes Hopf degree theorem to nontrivial bundles.

Survey on mean curvature flow with sphere theorems and Yau rigidity theory.

We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.

Green functions for GJMS operators on spheres derived, linking geometry and rigidity.

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…

We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty…

We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the $3$-Sphere and the $4$-Sphere' (arXiv:1706.01405), using the free mathematical software Sage.

Paper proves pinching theorem for minimal surfaces in spheres.

We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics.…

New $λ$-hypersurfaces not isometric to standard spheres.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

Modified proof constructs dual spheres for 4-manifolds.

The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.

The paper examines rigidity of special submanifolds in spheres with curvature constraints.

Characterizes simplicial complexes embedding into spheres with few vertices.

New theorem shows nearly spherical manifolds can be mapped from spheres.

Sphere theorems proved for manifolds with specific curvature conditions.

Simplified proof of Cerf's theorem on 3-sphere diffeomorphisms.

The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.

Sphere theorems extended to Riemannian foliations with new results on curvature and leaf spaces.

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 $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.

The paper constructs metrics on spheres with families of minimal hypersurfaces.

We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case of the main result of the first author in arxiv:1812.10454, and alr…

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…

Study eigenvalues of ellipsoids near a sphere, comparing to sphere's.

Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.

Sphere theorems for p-Laplacian eigenvalues established.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging to a certain class, then the diameter of $M$ does not exceed that of $\widetild…

3D spheres with certain properties approach the round sphere.

Extends Borsuk-Ulam theorem with applications in sphere coverings and colorings.

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.

We prove a quaternionic contact versions of the Obata's sphere theorems. We show that if the first positive eigenvalue of the sub-Laplacian on a compact qc manifold of dimension bigger than seven takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard…

Short proof of Strong Haken Theorem for 3-manifolds.

Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

This is a survey paper focusing on the interplay between the curvature and topology of a Riemannian manifold. The first part of the paper provides a background discussion, aimed at non-experts, of Hopf's pinching problem and the Sphere Theorem. In the second part, we sketch the proof of the Differentiable Sphere Theore…

New theorem links tropical phased matroids to higher-dimensional spheres.