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 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 : U ⊂ S 3 → 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.

Gauss maps of harmonic and minimal great circle fibrations

“…The map f is harmonic when we equip S 3 with the graphical metric g = g S 3 +f * g S 2 , where g S 3 is the metric of S 3 and g S 2 is the metric of S 2 . It seems that the methods developed in the present paper cannot be easily used, since in the structure equations (2.1) the curvature operator of the graphical metric shows up; for more information regarding this problem we refer to [17]. (e) In [19] it is shown that the graphical mean curvature flow (GMCF) will smoothly deform any strictly length decreasing map h : S 2 → S 2 into a constant map.…”

“…Consequently, g is a conformal map. This implies that the graph of g is a minimal surface in S 2 − × S 2 + ; see for example [2, Proposition 4.5.3], [7] or [17]. On the other hand, g is strictly length decreasing.…”