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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 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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$C^1$-regularity for local graph representations of immersions
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by Patrick Breuning PDF
Trans. Amer. Math. Soc. 365 (2013), 6185-6198 Request permission

Abstract:

We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius $r>0$. We show that for sufficiently small $C^0$-norm of the graph functions, each graph function is smooth with small $C^1$-norm.
References
  • P. Breuning: Immersions with local Lipschitz representation, dissertation, Freiburg, 2011.
  • P. Breuning: Immersions with bounded second fundamental form, preprint, 2011.
  • Kevin Corlette, Immersions with bounded curvature, Geom. Dedicata 33 (1990), no. 2, 153–161. MR 1050607, DOI 10.1007/BF00183081
  • Silvano Delladio, On hypersurfaces in $\textbf {R}^{n+1}$ with integral bounds on curvature, J. Geom. Anal. 11 (2001), no. 1, 17–42. MR 1829346, DOI 10.1007/BF02921952
  • Joel Langer, A compactness theorem for surfaces with $L_p$-bounded second fundamental form, Math. Ann. 270 (1985), no. 2, 223–234. MR 771980, DOI 10.1007/BF01456183
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Additional Information
  • Patrick Breuning
  • Affiliation: Institut für Mathematik der Goethe Universität Frankfurt am Main, Robert-Mayer-Straße 10, D-60325 Frankfurt am Main, Germany
  • Address at time of publication: Fakultät für Mathematik des Karlsruhe Institute of Technology, Institut für Analysis, Kaiserstrasse 89-93, D-76133 Karlsruhe, Germany
  • Email: breuning@math.uni-frankfurt.de, patrick.breuning@kit.edu
  • Received by editor(s): July 30, 2011
  • Published electronically: July 10, 2013
  • Additional Notes: This work was supported by the DFG-Forschergruppe Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis. The contents of this paper were part of the author’s dissertation, which was written at Universität Freiburg, Germany
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6185-6198
  • MSC (2010): Primary 53C42, 53B25
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05872-2
  • MathSciNet review: 3105747