A Nash–Kuiper theorem for $C^{1,\frac{1}{5}-\delta}$ immersions of surfaces in 3 dimensions

Camillo De Lellis; Dominik Inauen; László Székelyhidi Jr.

doi:10.4171/rmi/1019

JournalsrmiVol. 34, No. 3pp. 1119–1152

A Nash–Kuiper theorem for $C^{1, \frac{1}{5} - δ}$ immersions of surfaces in 3 dimensions

Camillo De Lellis
Universität Zürich, Switzerland
Dominik Inauen
Universität Zürich, Switzerland
László Székelyhidi Jr.
Universität Leipzig, Germany

$A Nash–Kuiper theorem for $C^{1,\frac{1}{5}-\delta}$ immersions of surfaces in 3 dimensions cover$

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Abstract

We prove that, given a $C^{2}$ Riemannian metric $g$ on the 2-dimensional disk $D_{2}$ , any short $C^{1}$ immersion of $(D_{2}, g)$ into $R^{3}$ can be uniformly approximated with $C^{1, α}$ isometric immersions for any $α < \frac{1}{5}$ . This statement improves previous results by Yu. F. Borisov and of a joint paper of the first and third author with S. Conti.

Cite this article

Camillo De Lellis, Dominik Inauen, László Székelyhidi Jr., A Nash–Kuiper theorem for $C^{1, \frac{1}{5} - δ}$ immersions of surfaces in 3 dimensions. Rev. Mat. Iberoam. 34 (2018), no. 3, pp. 1119–1152

DOI 10.4171/RMI/1019