Width estimate and doubly warped product
arXiv:2003.01315
Abstract
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 -manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be 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 -spheres with positive sectional curvatures.
We include an improvement for our main theorem under Ricci curvature lower bound in the last section. To appear on TAMS