论文标题
宽度估算和双重扭曲产品
Width estimate and doubly warped product
论文作者
论文摘要
在本文中，我们通过建立一个最佳的Lipschitz下限，以在可定向的$ 3 $ 3 $ manifolds上建立最佳的Lipschitz下限，以对Gromov的猜想（[3，猜想E]）提供肯定的答案。对于刚性，我们表明，如果达到最佳界限，则给定歧管的通用覆盖必须为$ \ mathbf r^2 \ times（-c，c）$，并具有一些双重扭曲的产品指标。这给出了具有正恒定曲率的双扭曲产品指标的表征。作为推论，我们还以$ 3 $ spheres的融合曲线获得了一个带正面曲线的浸入式圆锥形的焦点半径估计。
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.
