Harmonic Hopf constructions between spheres II

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-...

Harmonic Hopf Constructions between Spheres

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, II

1. A doomed-but-plausible trial calculation 2. Matrix exponentiation 3. Rotation-invariant derivatives on R 4. Instrinsic derivatives 5. Aside: GLn(R)-invariance of the Euler operator 6. ∑ i<j X 2 ij in coordinates on R 7. X` commutes with ∑ i<j X 2 ij 8. Stabilization of Hd by Xγ 9. Integration by parts for Xγ 10. Sup norms of derivatives of harmonic polynomials 11. Termwise differentiation of...

Harmonic maps between three-spheres

It is shown that smooth maps f : S → S contain two countable families of harmonic representatives in the homotopy classes of degree zero and one.