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arXiv 0905.0370math.DGmath.AP
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$C^{1,α}$等距嵌入的h-原理与刚性

h-Principle and Rigidity for $C^{1,α}$ Isometric Embeddings

Sergio Conti, Camillo De Lellis, László Székelyhidi

备注：
30 pages, 2 figures
发表信息：
Abel Symposia, vol. 7, Springer Berlin Heidelberg, 2012, pp. 83-116
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摘要：

本文研究了低余维黎曼流形的嵌入问题。著名的Nash和Kuiper的结果说，任何在余维1中的短嵌入都可以用$C^1$等距嵌入一致地近似。由于外尔问题中的经典刚性，这种说法对于一般的$C^2$嵌入显然不成立。事实上，Borisov将后者扩展到了类$C^{1,α}$与$α>2/3$的嵌入。另一方面，他宣布在纳什-柯伊伯声明可以扩展到当地的$C^{1,α}$嵌入与$α<(1+n+n^2)^{-1}$，其中$n$是维度的流形，提供的度量是分析。随后，二维情况的证明出现了。本文对一般维数和一般度量给出了所有这些命题的解析证明。

英文摘要：

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This statement clearly cannot be true for $C^2$ 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 $α>2/3$. On the other hand he announced in that the Nash-Kuiper statement can be extended to local $C^{1,α}$ embeddings with $α<(1+n+n^2)^{-1}$, where $n$ is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared. In this paper we provide analytic proofs of all these statements, for general dimension and general metric.

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