-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

DOI:
https://doi.org/10.1090/S0002-9947-06-04092-X

Published electronically:
May 17, 2006

MathSciNet review:
2231382

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-06-04092-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