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H-Principle and rigidity for C^1,α isometric embeddings


Conti, Sergio; De Lellis, Camillo (2010). H-Principle and rigidity for C^1,α isometric embeddings. In: Holgen, H; Karlsen, K H. Nonlinear differenial equations : the Abel symposium 2010. Berlin: Springer, 83-116.

Abstract

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60: 383-396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58: 545-556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58: 683-689, 1955) says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,α with α > 2/3 in (Borisov in Vestn. Leningr. Univ. 14(13): 20-26, 1959; Borisov in Vestn. Leningr. Univ. 15(19): 127-129, 1960). On the other hand he announced in (Borisov in Doklady 163: 869-871, 1965) that the Nash-Kuiper statement can be extended to local C1,α embeddings with α < (1 + n + n2)-1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared in (Borisov in Sib. Mat. Zh. 45(1): 25-61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.

Abstract

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60: 383-396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58: 545-556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58: 683-689, 1955) says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,α with α > 2/3 in (Borisov in Vestn. Leningr. Univ. 14(13): 20-26, 1959; Borisov in Vestn. Leningr. Univ. 15(19): 127-129, 1960). On the other hand he announced in (Borisov in Doklady 163: 869-871, 1965) that the Nash-Kuiper statement can be extended to local C1,α embeddings with α < (1 + n + n2)-1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared in (Borisov in Sib. Mat. Zh. 45(1): 25-61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.

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Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2010
Deposited On:15 Nov 2013 09:38
Last Modified:05 Apr 2016 17:09
Publisher:Springer
Series Name:Abel Symposia
Number:7
ISSN:2193-2808
ISBN:978-3-642-25360-7 (P) 978-3-642-25361-4 (E)
Publisher DOI:https://doi.org/10.1007/978-3-642-25361-4_5
Related URLs:http://link.springer.com/book/10.1007/978-3-642-25361-4 (Publisher)

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