Width estimate and doubly warped product
By Jintian Zhu. 2020
75 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Computational Mechanics, Computer Graphics and Computer-Aided Design
View Paper: https://www.semanticscholar.org/paper/7e36f1fcf714413b253bb6162964672ec64e31ba
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In Applied Mathematics
In Computational Mechanics
In Computer Graphics and Computer-Aided Design
Citations in the corpus
- Generalized soap bubbles and the topology of manifolds with positive scalar curvature (2020, methodology): “Recently, this approach was used by Gromov to obtain a list of comparison theorems in scalar curvature [13, 15], and later expanded in work of J. Zhu [43, 42].”