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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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Curvature estimates for surfaces with bounded mean curvature
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by Theodora Bourni and Giuseppe Tinaglia PDF
Trans. Amer. Math. Soc. 364 (2012), 5813-5828 Request permission


Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces in $\mathbb {R}^3$. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.
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Additional Information
  • Theodora Bourni
  • Affiliation: Freie Universität Berlin, Fachbereich Mathematik und Informatik, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany
  • Email:
  • Giuseppe Tinaglia
  • Affiliation: Mathematics Department, King’s College London, The Strand I, London WC2R 2LS, United Kingdom
  • Email:
  • Received by editor(s): July 16, 2010
  • Received by editor(s) in revised form: October 18, 2010
  • Published electronically: June 22, 2012
  • Additional Notes: The second author was partially supported by The Leverhulme Trust and EPSRC grant no. EP/I01294X/1
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 5813-5828
  • MSC (2010): Primary 53A10
  • DOI:
  • MathSciNet review: 2946933