arXiv research

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

We construct homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere with arbitrarily small k-dilation for each k greater than (m + 1)/2. We prove that homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere cannot have arbitrarily small k-dilation for k less than or equal …

In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Lege…

Study on minimal hypersurfaces in a unit sphere, proving specific isometries.

Study finds a minimum volume for vector fields on a punctured sphere.

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we construct unit $n$-sphere fibrations in $\mathbb{R}^{2n+1}$.

Study $4$-dim hypersurfaces with constant mean curvature in unit spheres.

Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.

It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to…

New convexity concept applied to sphere yields quermassintegral inequalities.

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $σ\in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.

Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.

The present paper discusses that a prescribed Gauss-Kronecker curvature problem on the product of unit spheres.

New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.

Totally geodesic hypersurfaces in a sphere have small total curvature.

Paper improves volume gap between minimal submanifolds and unit spheres.

Characterizes magnetic unit vector fields on Lie groups.

Minimal surfaces in S3(2) linked to vector fields on punctured sphere.

Analyzes convex structures in Teichmüller space unit tangent spheres.

Improved lower bound for the first eigenvalue of embedded minimal hypersurfaces in the unit sphere

Neural networks approximate unit spheres as polytopes.

Authors create stable proper biharmonic maps from unit ball to spheres.

The paper confirms conjectures about the topology of triangulated polyhedra and geodesic triangulations on spheres.

The study finds bounds for the first eigenvalue of the p-Laplacian on submanifolds.

Sharp lower bound found for area of vector fields on spherical annuli.

A curve of minimum length to enclose a unit sphere in 3D is at least 4π.

Constructs flows of tori in sphere perturbations for Morse homology.

Proves spheres with bounded curvatures must contain a unit ball.

In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincaré indexes around their singularities.

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that the intersection of A and B is H and the union is G and such that the complementary graphs A,B are both d-balls. The graph theoretic de…

Complete gradient Einstein-type Sasakian manifolds with α=0 are trivial or isometric to the unit sphere.

Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant…

3D spheres with certain properties approach the round sphere.

Paper confirms Yau's conjecture about sphere eigenvalues.

We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.

Unique symplectic fillings of odd spheres' cotangent bundles proven.

An isometric immersion $x:M^n\rightarrow S^{n+p}$ is called Willmore if it is an extremal submanifold of the Willmore functional: $W(x)=\int_{M^n} (S-nH^2)^{\frac{n}{2}}dv$, where $S$ is the norm square of the second fundamental form and $H$ is the mean curvature. Examples of Willmore submanifolds in the unit sphere ar…

Minimal normal curvature immersions in the unit ball studied.

Dual spherical conchoidal motion has been defined by Yapar. In this work, we define this motion on a dual hyperbolic unit sphere in the dual Lorentzian space with dual signature, and the results carried to the Lorentzian lines space by means of the Study s mapping. We also obtain the study maps of the orbits drawn on t…

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and…

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…

In this paper, we provide a complete classification for all the isometric cohomogeneity one actions on unit spheres. Using this theory, we can very easily classify all the isometric cohomogeneity one actions on the Riemannian symmetric spaces $\mathbb{C}\mathrm{P}^{m-1}$ and $\mathbb{H}\mathrm{P}^{m-1}$.

Study minimal networks on spheres and balls near standard metrics.

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geod…

We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, …

Functional determinant for mixed signature sphere products depends on sphere dimensions and parity.

New tilings of the 2-sphere from convex polyhedra in 3-sphere.