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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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$\mathbf {h}$-principles for hypersurfaces with prescribed principle curvatures and directions
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by Mohammad Ghomi and Marek Kossowski PDF
Trans. Amer. Math. Soc. 358 (2006), 4379-4393 Request permission

Abstract:

We prove that any compact orientable hypersurface with boundary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Further we construct regular homotopies (resp. isotopies) which control the principal curvatures and directions of hypersurfaces in a variety of ways. These results, which we prove by holonomic approximation, establish some h-principles in the sense of Gromov, and generalize theorems of Gluck and Pan on embedding and knotting of positively curved surfaces in 3-space.
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Additional Information
  • Mohammad Ghomi
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
  • MR Author ID: 687341
  • Email: ghomi@math.gatech.edu
  • Marek Kossowski
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • Email: kossowski@math.sc.edu
  • Received by editor(s): August 13, 2004
  • Published electronically: May 17, 2006
  • Additional Notes: The research of the first author was supported in part by NSF grant DMS-0204190 and CAREER award DMS-0332333.
  • © Copyright 2006 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 4379-4393
  • MSC (2000): Primary 53A07, 53C42; Secondary 57R42, 58J99
  • DOI: https://doi.org/10.1090/S0002-9947-06-04092-X
  • MathSciNet review: 2231382