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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ \mathbf{h}$-principles for hypersurfaces with prescribed principle curvatures and directions


Authors: Mohammad Ghomi and Marek Kossowski
Journal: Trans. Amer. Math. Soc. 358 (2006), 4379-4393
MSC (2000): Primary 53A07, 53C42; Secondary 57R42, 58J99
Published electronically: May 17, 2006
MathSciNet review: 2231382
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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
Email: ghomi@math.gatech.edu

Marek Kossowski
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: kossowski@math.sc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04092-X
PII: S 0002-9947(06)04092-X
Keywords: h-principle, regular homotopy, principal curvature, principal direction, Gauss curvature, hypersurface, Monge-Amp\`ere equation, jets and holonomy, holonomic approximation, immersion, embedding.
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.
Article copyright: © Copyright 2006 American Mathematical Society