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 -principles for hypersurfaces with prescribed principle curvatures and directions
Author(s):
Mohammad
Ghomi;
Marek
Kossowski
Journal:
Trans. Amer. Math. Soc.
358
(2006),
4379-4393.
MSC (2000):
Primary 53A07, 53C42;
Secondary 57R42, 58J99
Posted:
May 17, 2006
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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.
References:
-
- 1.
- S. Alexander and M. Ghomi.
The convex hull property and topology of hypersurfaces with nonnegative curvature.
Adv. Math., 180(1):324-354, 2003. MR 2019227 (2004k:53006) - 2.
- Y. Eliashberg and N. Mishachev.
Introduction to the  -principle.
American Mathematical Society, Providence, RI, 2002. MR 1909245 (2003g:53164) - 3.
- E. Feldman.
Nondegenerate curves on a Riemannian manifold.
J. Differential Geometry 5:187-210, 1971. MR 0291986 (45:1074) - 4.
- E. Feldman.
Immersions with nowhere vanishing mean curvature vector.
Topology, 12:210-227, 1973. MR 0326754 (48:5097) - 5.
- H. Geiges.
 -principles and flexibility in geometry.
Mem. Amer. Math. Soc., 164, no. 779, 2003. MR 1982875 (2004e:57031) - 6.
- M. Ghomi.
Strictly convex submanifolds and hypersurfaces of positive curvature.
J. Differential Geom., 57:239-271, 2001. MR 1879227 (2002k:52001) - 7.
- H. Gluck and L.-H. Pan.
Embedding and knotting of positive curvature surfaces in  -space.
Topology, 37(4):851-873, 1998. MR 1607752 (98m:57008) - 8.
- M. Golubitsky and V. Guillemin. Stable mappings and their singularities. Springer-Verlag, New York, Heidelberg, 1973. MR 0341518 (49:6269)
- 9.
- M. Gromov.
Partial differential relations.
Springer-Verlag, Berlin, 1986. MR 0864505 (90a:58201) - 10.
- B. Guan and J. Spruck.
Boundary value problem on  for surfaces of constant Gauss curvature.
Ann. of Math., 138:601-624, 1993. MR 1247995 (94i:53039) - 11.
- B. Guan and J. Spruck.
The existence of hypersurfaces of constant Gauss curvature with prescribed boundary.
J. Differential Geometry, 62:259-287, 2002. MR 1988505 (2004e:58019) - 12.
- L. Hauswirth.
Bridge principle for constant and positive Gauss curvature surfaces.
Comm. Anal. Geom., 7(3):497-550, 1999. MR 1698387 (2000i:58023) - 13.
- J. Little.
Nondegenerate homotopies of curves on the unit  -sphere.
J. Differential Geometry, 4:339-348, 1970. MR 0275333 (43:1090) - 14.
- J. Little.
Third order nondegenerate homotopies of space curves.
J. Differential Geometry, 5:503-515, 1971. MR 0291996 (45:1084) - 15.
- J. Little.
Space curves with positive torsion.
Ann. Mat. Pura Appl., 116(4):57-86, 1978. MR 0506974 (80k:53002) - 16.
- P. Røgen,
Embedding and knotting of flat compact surfaces in 3-space.
Comment. Math. Helv., 76(4):589-606, 2001. MR 1881699 (2002k:53007) - 17.
- N. S. Trudinger and X. Wang.
On locally convex hypersurfaces with boundary.
J. Reine Angew. Math., 551:11-32, 2002. MR 1932171 (2004b:58017)
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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:
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
Posted:
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 of article:
Copyright
2006,
American Mathematical Society
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