The Nash-Kuiper process for curves

Vincent Borrelli[1]; Saïd Jabrane[1]; Francis Lazarus[2]; Boris Thibert[3]

  • [1] Institut Camille Jordan, Université Lyon I, Villeurbanne, France
  • [2] CNRS, GIPSA-Lab, Université de Grenoble, France
  • [3] Laboratoire Jean Kuntzmann, Université de Grenoble, France

Séminaire de théorie spectrale et géométrie (2011-2012)

  • Volume: 30, page 1-19
  • ISSN: 1624-5458

Abstract

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A strictly short embedding is an embedding of a Riemannian manifold into an Euclidean space that strictly shortens distances. From such an embedding, the Nash-Kuiper process builds a sequence of maps converging toward an isometric embedding. In that paper, we describe this Nash-Kuiper process in the case of curves. We state an explicit formula for the limit normal map and perform its Fourier series expansion. We then adress the question of Holder regularity of the limit map.

How to cite

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Borrelli, Vincent, et al. "The Nash-Kuiper process for curves." Séminaire de théorie spectrale et géométrie 30 (2011-2012): 1-19. <http://eudml.org/doc/275683>.

@article{Borrelli2011-2012,
abstract = {A strictly short embedding is an embedding of a Riemannian manifold into an Euclidean space that strictly shortens distances. From such an embedding, the Nash-Kuiper process builds a sequence of maps converging toward an isometric embedding. In that paper, we describe this Nash-Kuiper process in the case of curves. We state an explicit formula for the limit normal map and perform its Fourier series expansion. We then adress the question of Holder regularity of the limit map.},
affiliation = {Institut Camille Jordan, Université Lyon I, Villeurbanne, France; Institut Camille Jordan, Université Lyon I, Villeurbanne, France; CNRS, GIPSA-Lab, Université de Grenoble, France; Laboratoire Jean Kuntzmann, Université de Grenoble, France},
author = {Borrelli, Vincent, Jabrane, Saïd, Lazarus, Francis, Thibert, Boris},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {convex integration; isometric embedding Riesz product; isometric embeddings; convex integration theory; barycentric formula},
language = {eng},
pages = {1-19},
publisher = {Institut Fourier},
title = {The Nash-Kuiper process for curves},
url = {http://eudml.org/doc/275683},
volume = {30},
year = {2011-2012},
}

TY - JOUR
AU - Borrelli, Vincent
AU - Jabrane, Saïd
AU - Lazarus, Francis
AU - Thibert, Boris
TI - The Nash-Kuiper process for curves
JO - Séminaire de théorie spectrale et géométrie
PY - 2011-2012
PB - Institut Fourier
VL - 30
SP - 1
EP - 19
AB - A strictly short embedding is an embedding of a Riemannian manifold into an Euclidean space that strictly shortens distances. From such an embedding, the Nash-Kuiper process builds a sequence of maps converging toward an isometric embedding. In that paper, we describe this Nash-Kuiper process in the case of curves. We state an explicit formula for the limit normal map and perform its Fourier series expansion. We then adress the question of Holder regularity of the limit map.
LA - eng
KW - convex integration; isometric embedding Riesz product; isometric embeddings; convex integration theory; barycentric formula
UR - http://eudml.org/doc/275683
ER -

References

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  12. J.-P. Kahane, Jacques Peyrière et les produits de Riesz 
  13. N. Kuiper, On C 1 -isometric imbeddings, Indag. Math. 17 (1955), 545-556 Zbl0067.39601MR75640
  14. J. Nash, C 1 -isometric imbeddings, Ann. of Math. (2) 60 (1954), 383-396 Zbl0058.37703MR65993
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