Rigidity of the Hopf fibration

Abstract

In this paper, we study minimal maps between euclidean spheres. The Hopf fibrations provide explicit examples of such minimal maps. Moreover, their corresponding graphs have second fundamental form of constant norm. We prove that a minimal submersion from $S^3$ to $S^2$ whose Gauss map satisfies a suitable pinching condition must be weakly conformal and with totally geodesic fibers. As a consequence, we obtain that an equivariant minimal submersion from $S^3$ to $S^2$ coincides with the Hopf fibration. Furthermore, we prove that a minimal map $f:S^3 \to S^2$ with constant singular values and constant norm of the second fundamental form is either constant or, up to isometries, coincides with the Hopf fibration.