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

TL;DR

This paper investigates harmonic maps from the 3-sphere to the 2-sphere, providing evidence for Eells' conjecture that such maps are essentially Hopf fibrations, under certain geometric conditions, and establishes related rigidity results.

Contribution

It proves the conjecture under specific Hessian and singular value conditions, extending rigidity results similar to classical pinching theorems for minimal hypersurfaces.

Findings

Validation of Eells' conjecture under certain conditions

A new pinching theorem for harmonic maps from $S^3$ to $S^2$

Rigidity results for 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.