Harmonic analysis on spheres, II

Harmonic and Quasi-Harmonic Spheres, Part II

This is in the sequel of our previous work [LW] on the study of the approximated harmonic maps in high dimensions. The main purpose of the present article is to understand the bubbling phenomena as well as the energy quantization beyond the natural conformal dimension two for the Dirichelet integral. This will be important toward our understandings of the defect measures and the energy concentr...

Harmonic analysis on spheres

1. Calculus on spheres 2. Spherical Laplacian from Euclidean 3. Eigenvectors for the spherical Laplacian 4. Invariant integrals on spheres 5. L spectral decompositions on spheres 6. Sup-norms of spherical harmonics on Sn−1 7. Pointwise convergence of Fourier-Laplace series 8. Irreducibility of representation spaces for O(n) 9. Hecke’s identity • Appendix: Bernstein’s proof of Weierstraß approxi...

Harmonic Hopf Constructions between Spheres Ii

This paper can be seen as the final remark of the previous paper written by the first author [D]. We consider the existence of harmonic maps between two spheres, via Hopf constructions. Given a non trivial bi-eigenmap f : S × S −→ S with bi-eigenvalue (λ, μ) (λ, μ > 0) and a continuous function α : [ 0, 2 ]−→ [ 0, π ] with α(0) = 0 , α(2 ) = π , one defines a map u: S p+q+1 −→ S , called the α-...

Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields

We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space ΩG of a compact Lie group G a...

Approximation Theory and Harmonic Analysis on Spheres and Balls