  In this paper, we give an affirmative answer to Gromov's conjecture ([3,
Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of
smooth functions on orientable open $3$-manifolds with uniformly positive
sectional curvatures. For rigidity we show that the universal covering of the
given manifold must be $\mathbf R^2\times (-c,c)$ with some doubly warped
product metric if the optimal bound is attained. This gives a characterization
for doubly warped product metrics with positive constant curvature. As a
corollary, we also obtain a focal radius estimate for immersed toruses in
$3$-spheres with positive sectional curvatures.

Differential Geometry