  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.
