Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Published electronically: November 25, 2020
MathSciNet review: 4196400
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 53C21, 53C24

Retrieve articles in all journals with MSC (2020): 53C21, 53C24

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
MR Author ID: 1321060

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