Sphere Sm Research Articles

Concentration on submanifolds of positively curved homogeneous spaces

A classical result of Milman roughly states that every Lipschitz function on Sn is almost constant on a sufficiently high-dimensional sphere Sm⊂Sn. In this paper we extend the result by proving that any Lipschitz function on a positively curved homogeneous space is almost constant on a high dimensional submanifold.

Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere Sm(c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.

A note on the equivariant cobordism of generalized Dold manifolds

Let (X,J) be an almost complex manifold with a (smooth) involution σ:X→X such that Fix(σ)≠∅. Assume that σ is a complex conjugation, i.e., the differential of σ anti-commutes with J. The space P(m,X):=Sm×X/∼ where (v,x)∼(−v,σ(x)) is known as a generalized Dold manifold. Suppose that a group G≅Z2s acts smoothly on X such that g∘σ=σ∘g for all g∈G. Using the action of the diagonal subgroup D=O(1)m+1⊂O(m+1) on the sphere Sm, we obtain an action of G=D×G on Sm×X, which descends to a (smooth) action of G on P(m,X). When the stationary point set XG for the G action on X is finite, the same also holds for the G action on P(m,X). The main result of this note is that the equivariant cobordism class [P(m,X),G] vanishes if and only if [X,G] vanishes. We illustrate this result in the case when X is the complex flag manifold, σ is the natural complex conjugation and G≅(Z2)n is contained in the diagonal subgroup of U(n).

Ricci curvature on warped product submanifolds in spheres with geometric applications

The goal of this paper is to construct a fundamental theorem for the Ricci curvature inequality via partially minimal isometric warped product immersions into an m-dimensional unit sphere Sm, involving the Laplacian of a well defined warping function, the squared norm of a warping function and the squared norm of the mean curvature. Moreover, the equality cases are discussed in detail and some applications are also derived due to involvement of the warping function. As applications, we provide sufficient condition that the base N1p is isometric to the sphere Sp(λ1p) with constant sectional curvature c=λ1p. The obtained results in the paper give the partial solution of Ricci curvature conjecture, also known as Chen-Ricci inequality obtained by Chen (1999).

Branched Coverings of Manifolds and nH-Spaces

We show that on the sphere Sm, m ≠ 1, 3, 7, there exists an nm-valued multiplication with unit for some nm ∈ {2, 4, 8}. We also explicitly construct a 2k−1-fold branched covering of $$S^{m_1}\;\times\cdots\times\;S^{m_k}$$ the product Sm1× ··· × Smk of k spheres over the sphere Sm, m = m1 + ··· + mk.

On spherical submanifolds with finite type spherical Gauss map

Abstract Chen and Lue (2007) initiated the study of spherical submanifolds with finite type spherical Gauss map. In this paper, we firstly prove that a submanifold Mn of the unit sphere Sm−1 has non-mass-symmetric 1-type spherical Gauss map if and only if Mn is an open part of a small n-sphere of a totally geodesic (n + 1)- sphere Sn+1 . Sm−1. Thenwe show that a non-totally umbilical hypersurfaceM of Sn+1 with nonzero constant mean curvature in Sn+1 has mass-symmetric 2-type spherical Gauss map if and only if the scalar curvature curvature of M is constant. Finally, we classify constant mean curvature surfaces in S3 with mass-symmetric 2-type spherical Gauss map.

Functions on the sphere with critical points in pairs and orthogonal geodesic chords

Using an estimate on the number of critical points for a Morse-even function on the sphere Sm, m≥1, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.

Degenerate solutions of a nonlinear elliptic equation on the sphere

We consider the equation Δu=λ(u−uq) on the round sphere Sm, m≥3, with λ>0 and 1<q<m+2m−2. By studying the local structure of solutions bifurcating from the constant solution we show that for any isoparametric hypersurface M in the sphere there is a value of λ and a solution of the equation at which the linearized equation has nontrivial solutions. At the same time we obtain new solutions of the equation.

Weak conformality of stable stationary maps for a functional related to conformality

Let (M,g), (N,h) be compact Riemannian manifolds without boundary, and let f be a smooth map from M into N. We consider a covariant symmetric tensor Tf=f⁎h−1m‖df‖2g, where f⁎h denotes the pullback of the metric h by f, and m is the dimension of the manifold M. The tensor Tf vanishes if and only if the map f is weakly conformal. The norm ‖Tf‖ is a quantity which is a measure of conformality of f at each point. In Nakauchi (2011) [5] the second author introduced maps which are critical points of the functional Φ(f)=∫M‖Tf‖2dvg. We call such maps C-stationary maps. Any conformal map or more generally any weakly conformal map is a C-stationary map. It is of interest to find when a C-stationary map is a (weakly) conformal map.In this paper we prove the following result. If f is a stable C-stationary map from the standard sphere Sm(m⩾5) or into the standard sphere Sn(n⩾5), then f is a weakly conformal map.

New AdS solitons and brane worlds with compact extra-dimensions

We construct new static, asymptotically AdS solutions where the conformal infinity is the product of Minkowski spacetime $M_n$ and a sphere $S^m$. Both globally regular, soliton-type solutions and black hole solutions are considered. The black holes can be viewed as natural AdS generalizations of the Schwarzschild black branes in Kaluza-Klein theory. The solitons provide new brane-world models with compact extra-dimensions. Different from the Randall-Sundrum single-brane scenario, a Schwarzschild black hole on the Ricci flat part of these branes does not lead to a naked singularity in the bulk.

Coverage processes on spheres and condition numbers for linear programming

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let p(n, m, α) be the probability that n spherical caps of angular radius α in Sm do not cover the whole sphere Sm. We give an exact formula for p(n, m, α) in the case α∈[π/2, π] and an upper bound for p(n, m, α) in the case α∈[0, π/2] which tends to p(n, m, π/2) when α→π/2. In the case α∈[0, π/2] this yields upper bounds for the expected number of spherical caps of radius α that are needed to cover Sm. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem ∃x∈ℝm+1Ax≤0, x≠0 where A∈ℝn×(m+1) is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to A being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all n>m, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

A CLASSIFICATION OF HYPERSURFACES WITH PARALLEL PARA-BLASCHKE TENSOR IN Sm+1

In this paper, we classify all immersed hypersurfaces in the unit sphere Sm+1 with parallel para-Blaschke tensor.

The Möbius characterizations of Willmore tori and Veronese submanifolds in the unit sphere

Suppose M is a m-dimensional submanifold without umbilic points in the (m + p)-dimensional unit sphere Sm+p. Four basic invariants of Mm under the Mobius transformation group of Sm+p are a symmetric positive definite 2-form g called the Mobius metric, a section B of the normal bundle called the Mobius second fundamental form, a 1-form F called the Mobius form, and a symmetric (0,2) tensor A called the Blaschke tensor. In the Mobius geometry of submanifolds, the most important examples of Mobius minimal submanifolds (also called Willmore submanifolds) are Willmore tori and Veronese submanifolds. In this paper, several fundamental inequalities of the Mobius geometry of submanifolds are established and the Mobius characterizations of Willmore tori and Veronese submanifolds are presented by using Mobius invariants.

Eigenvalues of Integral Operators Defined by Smooth Positive Definite Kernels

We consider integral operators defined by positive definite kernels \(K : X \times X \rightarrow {\mathbb{C}}\), where X is a metric space endowed with a strictly-positive measure. We update upon connections between two concepts of positive definiteness and upgrade on results related to Mercer like kernels. Under smoothness assumptions on K, we present decay rates for the eigenvalues of the integral operator, employing adapted to our purposes multidimensional versions of known techniques used to analyze similar problems in the case where X is an interval. The results cover the case when X is a subset of \({\mathbb{R}}^{m}\) endowed with the induced Lebesgue measure and the case when X is a subset of the sphere Sm endowed with the induced surface Lebesgue measure.

Isoperimetry of Waists and Concentration of Maps

Let f : Sn → Rk be a continuous map where Sn is the unit n-sphere. Then there exists a point z ∈ Rk such that the spherical n-volumes of the e-neighbours of the level Yz = f−1(z) ⊂ Sn, denoted Yz + e ⊂ Sn, satisfy Vol(Yz + e) ≥ Vol(Sn−k + e) ( )Sn for all e > 0, where Sn−k ⊂ Sn denotes an equatorial (n− k)-sphere. This is proven in §5.9. Remarks. (a) If k = n, and card f−1(z) ≤ 2, z ∈ Rn, then ( )Sn applied to e = π/2 amounts to the Borsuk–Ulam theorem: some level f−1(z) of f : Sn → Rk equals a pair of opposite points. Not surprisingly, our argument in the general case depends on Z2-cohomological considerations. (b) If k = 1 one may take the Levy mean of f for z ∈ R, where the level f−1(z) ⊂ Sn divides the sphere into equal halves (i.e. where Vol(f−1(−∞, z]) and Vol(f−1[z,∞)) are both ≥ 12 Vol(Sn)). Then ( )SN follows from the spherical isoperimetric inequality (see 9.2.B). (c) The inequality ( )Sn for e → 0 shows that the Minkowski m-volume of f−1(z) for m = n − k is ≥ than that of the equatorial sphere Sm ⊂ Sn. Yet, it remains unclear if some level of f has the Hausdorff measure ≥ Volm(S). (If f is a generic smooth map, a level with Haumesm ≥ Volm(S) is delivered by the Almgren–Morse theory, see [Gr5].)

Selfcoincidences and roots in Nielsen theory

Given two maps f1 and f2 from the sphere Sm to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On the one hand, the resulting bordism class of coincidence data and the corresponding Nielsen numbers are strong looseness obstructions. On the other hand, the values which these invariants may possibly assume turn out to satisfy severe restrictions, e.g. the Nielsen numbers can only take the values 0, 1 or the cardinality of the fundamental group of N. In order to show this we compare different Nielsen classes in the root case (where f1 or f2 is constant) and we use the fact that all but possibly one Nielsen class are inessential in the selfcoincidence case (where f1 = f2). We also deduce strong vanishing results.

Zonal decompositions for spherical monogenics

AbstractIn this paper, a specific type of eigenfunctions for the Gamma operator Γ on the unit sphere Sm−1 in ℝm is constructed within the context of Clifford analysis, by means of a recursive argument. This leads to zonal decompositions for the spherical monogenics. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Möbius Characterization of Some Submanifolds in the Unit Sphere

Let M m be an m-dimensional umbilic-free submanifold in the m + p-dimensional unit sphere Sm+p. Three basic invariants of M m under the Mobius transformation group of S m are a 1-form Φ, called Mobius form, a symmetric (0,2) tensor A, called Blaschke tensor, and a positive definite (0,2) tensor g, called Mobius metric. We denote the Mobius scalar curvature by R and the trace-free Blaschke tensor by \( \tilde{{\bf A}}: = {\bf A} - \frac{1}{m}\hbox{tr}({{\bf A}}){\bf {g}} \). In this paper, we prove a local classification result under the assumption of vanishing Mobius form and an inequality of the type $$ c_{1} \left\| {\left. \tilde{{\bf A}}\right\| \le R - c_{2} } \right. , $$ where c1, c2 are appropriate real constants, c1 depending on the dimension, and c2 on the dimension and codimension.

Hyperbolic L 2–modules with Reproducing Kernels

In this paper, the Dirac operator on the Klein model for the hyperbolic space is considered. A function space containing L2–functions on the sphere Sm−1 in ℝm, which are boundary values of solutions for this operator, is defined, and it is proved that this gives rise to a Hilbert module with a reproducing kernel.

M�bius geometry of hypersurfaces with constant mean curvature and scalar curvature

Let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface in the (m+1)-dimensional unit sphere Sm+1 without umbilics. Four basic invariants of x under the Mobius transformation group in Sm+1 are a Riemannian metric g called Mobius metric, a 1-form Φ called Mobius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Mobius second fundamental form. In this paper, we prove the following classification theorem: let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface, which satisfies (i) Φ≡0, (ii) A+λg+μB≡0 for some functions λ and μ, then λ and μ must be constant, and x is Mobius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in Sm+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space Rm+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space Hm+1. This result shows that one can use Mobius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in Sm+1, Rm+1 and Hm+1.