Mapping a sphere into Euclidean space

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 3–8, September, 1989.     

Lipschitz maps and nets in Euclidean space

((Without abstract)) Submitted: April 1997, Final version: July 1997     

Harmonic Gauss maps of submanifolds of arbitrary codimension of the Euclidean space and sphere and some applications

It is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too.     

Partition of Euclidean space into polyhedra induced by a small deformation of the densest lattice sphere packing

The number of combinatorially nonequivalent Dirichlet-Voronoi diagrams constructed for the centers of balls in the packing obtained from the densest lattice packing of equal spheres by a small displacement of the spheres is estimated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 79–84, 2005.     

Properties of continuous functions on a normed space and its sphere

Let K ⊂ ℝ³ be a convex body of unit volume. It is proved that K contains an affine-regular pentagonal prism of volume 4( 5 - 2?5 ) \mathord

Equivariant harmonic maps into the sphere via isoparametric maps

By using concrete isoparametric maps we obtain some new equivariant harmonic maps between spheres and solve equivariant boundary value problems for harmonic maps from unit open ballB ^m+1 intoS ⁿ. Research partially supported by NNSFC, SFECC and ICTP.     

On a mapping of a projective space into a sphere

We obtain an exact estimate for the minimum multiplicity of a continuous finite-to-one mapping of a projective space into a sphere for all dimensions. For finite-to-one mappings of a projective space into a Euclidean space, we obtain an exact estimate for this multiplicity for n = 2, 3. For n ≥ 4, we prove that this estimate does not exceed 4. Several open questions are formulated.     

Geometry of finite-dimensional normed spaces,and continuous functions on the Euclidean sphere

Let ℝⁿ be the n-dimensional Euclidean space, and let { · } be a norm in Rⁿ. Two lines ℓ₁ and ℓ₂ in ℝⁿ are said to be { · }-orthogonal if their { · }-unit direction vectors e ₁ and e ₂ satisfy {e ₁ + e ₂} = {e ₁ − e ₂}. It is proved that for any two norms { · } and { · }′ in ℝⁿ there are n lines ℓ₁, ..., ℓ_n that are { · }-and { · }′-orthogonal simultaneously. Let be a continuous function on the unit sphere with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in , and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even, then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e ₁, ..., e _n such that for 1 ≤ i ≤ j ≤ n we have . Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117.     

Regularity of p-harmonic maps from the p-dimensional ball into a sphere

We prove that, forp≥2, all weaklyp-harmonic mapsu=(u ₁,...,u _n ) from thep-dimensional ball into a sphere, i.e. weak solutions of classW ^1,p of the constrained eliptic system $$\begin{gathered} - div(|\nabla u|^{p - 2} \nabla u_i ) = u_i |\nabla u|^p \hfill \\ \sum {(u_i )} ^2 = 1, \hfill \\ \end{gathered} $$ are Hölder continuous. This result is an analogue of an earlier theorem of F. Hélein for the casep=2.     

The Taikov functional in the space of algebraic polynomials on the multidimensional Euclidean sphere

We discuss three related extremal problems on the set of algebraic polynomials of given degree n on the unit sphere $ \mathbb{S}^{m - 1} $ of Euclidean space ? ^m of dimension m ≥ 2. (1) The norm of the functional F(h) = FhP _n = ∫_?(h) P _n (x)dx, which is equal to the integral over the spherical cap ?(h) of angular radius arccos h, ?1 < h < 1, on the set with the norm of the space L( $ \mathbb{S}^{m - 1} $ ) of summable functions on the sphere. (2) The best approximation in L _∞( $ \mathbb{S}^{m - 1} $ ) of the characteristic function χ _h of the cap ?(h) by the subspace of functions from L _∞( $ \mathbb{S}^{m - 1} $ ) that are orthogonal to the space of polynomials . (3) The best approximation in the space L( $ \mathbb{S}^{m - 1} $ ) of the function χ _h by the space of polynomials . We present the solution of all three problems for the value h = t(n,m) which is the largest root of the polynomial in a single variable of degree n + 1 least deviating from zero in the space L ₁ ^? on the interval (?1, 1) with ultraspheric weight ?(t) = (1 ? t ²) ^α , α = (m ? 3)/2.     

Analytic and harmonic maps into a topological space

The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.     

Some new symmetry results for elliptic problems on the sphere and in Euclidean space

We prove some new symmetry results for positive solutions of elliptic problems in ℝ ⁿ and on the sphere. The proofs are based on the moving plane method, rearrangement arguments and stereographic projection.