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

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.