Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration

Abstract

It was conjectured by Eells that the only harmonic maps $f:S^3\to S^2$ are Hopf fibrations composed with conformal maps of $S^2$. We support this conjecture by proving its validity under suitable conditions on the Hessian and the singular values of $f$. Among the results, we obtain a pinching theorem in the spirit of that of Simons, Lawson and Chern, do Carmo and Kobayashi for minimal hypersurfaces in the sphere.

Adjustments have been made to Theorem C, and additional details have been included in its proof. Parts of the former Theorem C now appear in a new Theorem D