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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Width estimate and doubly warped product
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by Jintian Zhu PDF
Trans. Amer. Math. Soc. 374 (2021), 1497-1511 Request permission


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
  • MR Author ID: 1321060
  • Email:,
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 1497-1511
  • MSC (2020): Primary 53C21; Secondary 53C24
  • DOI:
  • MathSciNet review: 4196400