arXiv research

The study characterizes round spheres in Euclidean space based on r-mean curvature conditions.

Existence and instability of biharmonic maps from balls to spheres.

Solves four problems related to sphere families in 3D space.

Conditions for Riemannian manifolds to be Euclidean spheres or spaces.

A minimal hypersurface in a sphere is uniquely determined.

Compact gradient ρ-Einstein solitons are isometric to Euclidean spheres.

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

Improved gap for mean curvature of biharmonic hypersurfaces in spheres.

Simple sphere eversion with a unique point.

The paper explores biconservative surfaces in a 4D sphere, finding a unique family of non-isometric surfaces.

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for m…

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2π}{3}$ but less than $2π$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result give…

A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe…

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

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

Study classifies equidistant decompositions in 2D spaces.

The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.

In the present study we consider knotted spheres in Euclidean $4$-space $\mathbb{E}^{4}$. Firstly, we give some basic curvature properties of knotted spheres in $\mathbb{E}^{4}$. Further, we obtained some results related with the conjugate nets and Laplace transforms of these kind of surfaces.

The study finds conditions for area-minimizing cones over submanifolds.

The study identifies surfaces with Maslovian normal bundles.

It is proved some results about existence and non existence of unit normal sections of submanifolds of the Euclidean space and sphere which associated Gauss maps are harmonic. Some applications to CMC hypersurfaces of the sphere and isoparametric submanifolds are obtained too.

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

We consider the Jacobi operator, defined on a closed oriented hypersurfaces immersed in the Euclidean space with the same volume of the unit sphere. We show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we prove that the first eigenvalue of the Ja…

A curve around a sphere must be at least 4π long.

Free boundary minimal submanifolds with boundaries on concentric spheres

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear e…

The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.

Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.

We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres.

The Green function on spheres in 3D implies the surface is a round sphere.

Proves isometric embeddings in Euclidean spaces for RCD spaces.

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generaliz…

We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations.

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

The aim of this paper is to present a complete description of all rotational linear Weingarten surface into the Euclidean sphere S3. These surfaces are characterized by a linear relation aH+bK=c, where H and K stand for their mean and Gaussian curvatures, respectively, whereas a; b and c are real constants.

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

We say that a topologically embedded 3-sphere in a smoothing of Euclidean 4-space is a barrier provided, roughly, no diffeomorphism of the 4-manifold moves the 3-sphere off itself. In this paper we construct infinitely many one parameter families of distinct smoothings of 4-space with barrier 3-spheres. \par The existe…

Paper proves no stable Yang-Mills fields on spheres.

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

We use an idea of Wang and Yau to give a new definition of quasi-local mass for a topological sphere in an initial date set. The new definition modifies Brown-York's definition by using certain spinor norm as lapse function. And it requires mean curvature of the topological sphere satisfies apparent horizon conditions,…

Study shows spheres in high dimensions have maximum volume if they are smooth and have a specific reach.

Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.

A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux…

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for …

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

The purpose of this paper is to give an effective construction for some induced structures on spheres or product of spheres of codimension 1, 2 or 3, respectively, in Euclidean space endowed with an almost product structure.

A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers $p$ for which a given homotopy sphere admits a special generic map into $\mathbb{R}^{p}$. By means of t…