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Transactions of the American Mathematical Society

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

 
 

 

Curvature estimates for surfaces with bounded mean curvature


Authors: Theodora Bourni and Giuseppe Tinaglia
Journal: Trans. Amer. Math. Soc. 364 (2012), 5813-5828
MSC (2010): Primary 53A10
DOI: https://doi.org/10.1090/S0002-9947-2012-05487-0
Published electronically: June 22, 2012
MathSciNet review: 2946933
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Abstract: 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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Theodora Bourni
Affiliation: Freie Universität Berlin, Fachbereich Mathematik und Informatik, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany
Email: bourni@math.fu-berlin.de

Giuseppe Tinaglia
Affiliation: Mathematics Department, King’s College London, The Strand I, London WC2R 2LS, United Kingdom
Email: giuseppe.tinaglia@kcl.ac.uk

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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.