Rigidity of the Hopf fibration

TL;DR

This paper investigates the rigidity properties of the Hopf fibration, showing under certain conditions that minimal maps from S^3 to S^2 must be equivalent to the Hopf fibration, highlighting its unique geometric structure.

Contribution

It establishes rigidity results for minimal submersions from S^3 to S^2, characterizing when they are necessarily the Hopf fibration or constant maps.

Findings

Any equivariant minimal submersion from S^3 to S^2 is the Hopf fibration.

Minimal maps with constant singular values are either constant or the Hopf fibration.

Such maps have fibers that are totally geodesic under certain conditions.

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.