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.

中文翻译：

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宽度估计和双翘产品

在本文中，我们通过为具有均匀正截面曲率的可定向开放 $3$-流形上的一类光滑函数建立最优 Lipschitz 下界，给出了对 Gromov 猜想（[3，猜想 E]）的肯定回答。为了刚性，我们证明给定流形的通用覆盖必须是 $\mathbf R^2\times (-c,c)$ 如果达到最佳边界，则具有一些双翘曲的乘积度量。这给出了具有正恒定曲率的双翘曲产品度量的特征。作为推论，我们还获得了具有正截面曲率的 $3$-球体中浸入环面的焦半径估计。