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arXiv · 2511.16522

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

Athanasios Georgakopoulos·Marco Magliaro·Luciano Mari·Andreas Savas-Halilaj
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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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Athanasios Georgakopoulos, Marco Magliaro, Luciano Mari, Andreas Savas-Halilaj. 2026-01-27. Harmonic maps from $S^3$ to $S^2$ and the rigidity of the Hopf fibration. https://arxiv.org/abs/2511.16522

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