$\mathbf {h}$-principles for hypersurfaces with prescribed principle curvatures and directions
HTML articles powered by AMS MathViewer
- 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.References
- Stephanie Alexander and Mohammad Ghomi, The convex hull property and topology of hypersurfaces with nonnegative curvature, Adv. Math. 180 (2003), no. 1, 324–354. MR 2019227, DOI 10.1016/S0001-8708(03)00006-9
- Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
- E. A. Feldman, Nondegenerate curves on a Riemannian manifold, J. Differential Geometry 5 (1971), 187–210. MR 291986, DOI 10.4310/jdg/1214429788
- E. A. Feldman, Immersions with nowhere vanishing mean curvature vector, Topology 12 (1973), 210–227. MR 326754, DOI 10.1016/0040-9383(73)90007-4
- Hansjörg Geiges, $h$-principles and flexibility in geometry, Mem. Amer. Math. Soc. 164 (2003), no. 779, viii+58. MR 1982875, DOI 10.1090/memo/0779
- Mohammad Ghomi, Strictly convex submanifolds and hypersurfaces of positive curvature, J. Differential Geom. 57 (2001), no. 2, 239–271. MR 1879227
- Herman Gluck and Liu-Hua Pan, Embedding and knotting of positive curvature surfaces in $3$-space, Topology 37 (1998), no. 4, 851–873. MR 1607752, DOI 10.1016/S0040-9383(97)00056-6
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518, DOI 10.1007/978-1-4615-7904-5
- Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
- Bo Guan and Joel Spruck, Boundary-value problems on $S^n$ for surfaces of constant Gauss curvature, Ann. of Math. (2) 138 (1993), no. 3, 601–624. MR 1247995, DOI 10.2307/2946558
- Bo Guan and Joel Spruck, The existence of hypersurfaces of constant Gauss curvature with prescribed boundary, J. Differential Geom. 62 (2002), no. 2, 259–287. MR 1988505
- Laurent Hauswirth, Bridge principle for constant and positive Gauss curvature surfaces, Comm. Anal. Geom. 7 (1999), no. 3, 497–550. MR 1698387, DOI 10.4310/CAG.1999.v7.n3.a2
- John A. Little, Nondegenerate homotopies of curves on the unit $2$-sphere, J. Differential Geometry 4 (1970), 339–348. MR 275333
- John A. Little, Third order nondegenerate homotopies of space curves, J. Differential Geometry 5 (1971), 503–515. MR 291996
- John A. Little, Space curves with positive torsion, Ann. Mat. Pura Appl. (4) 116 (1978), 57–86. MR 506974, DOI 10.1007/BF02413867
- Peter Røgen, Embedding and knotting of flat compact surfaces in 3-space, Comment. Math. Helv. 76 (2001), no. 4, 589–606. MR 1881699, DOI 10.1007/s00014-001-0000-6
- Neil S. Trudinger and Xu-Jia Wang, On locally convex hypersurfaces with boundary, J. Reine Angew. Math. 551 (2002), 11–32. MR 1932171, DOI 10.1515/crll.2002.078
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