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.
Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration / A. Georgakopoulos, M. Magliaro, L. Mari, A. Savas-Halilaj. - (2025 Nov 20).
Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration
M. Magliaro;L. Mari;
2025
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.
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