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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Width estimate and doubly warped product


Author: Jintian Zhu
Journal: Trans. Amer. Math. Soc. 374 (2021), 1497-1511
MSC (2020): Primary 53C21; Secondary 53C24
DOI: https://doi.org/10.1090/tran/8263
Published electronically: November 25, 2020
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Abstract: 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 $ \mathbf R^2\times (-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.


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Additional Information

Jintian Zhu
Affiliation: Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Email: zhujt@pku.edu.cn, jintian@uchicago.edu

DOI: https://doi.org/10.1090/tran/8263
Received by editor(s): May 10, 2020
Received by editor(s) in revised form: July 21, 2020
Published electronically: November 25, 2020
Additional Notes: This work was partially supported by China Scholarship Council and the NSCF grants No. 11671015 and 11731001.
Article copyright: © Copyright 2020 American Mathematical Society