Width estimate and doubly warped product

In this paper, we give an affirmative answer to Gromov’s conjecture (Geom. Funct. Anal. 28 (2018), pp. 645–726, Conjecture E) by establishing an optimal Lipschitz lower bound for a class of smooth functions on connected orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that if the optimal bound is attained the given manifold must be a quotient space of R²× (-c,c) with some doubly warped product metric. 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.