Revista Matemática Iberoamericana

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Volume 34, Issue 3, 2018, pp. 1119–1152
DOI: 10.4171/RMI/1019

Published online: 2018-08-27

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

Camillo De Lellis[1], Dominik Inauen[2] and László Székelyhidi Jr.[3]

(1) Universität Zürich, Switzerland
(2) Universität Zürich, Switzerland
(3) Universität Leipzig, Germany

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 $\mathbb R^3$ can be uniformly approximated with $C^{1,\alpha}$ isometric immersions for any $\alpha < \frac{1}{5}$. This statement improves previous results by Yu. F. Borisov and of a joint paper of the first and third author with S. Conti.

Keywords: Isometric embedding, convex integration, Nash–Kuiper

De Lellis Camillo, Inauen Dominik, Székelyhidi Jr. László: A Nash–Kuiper theorem for $C^{1,\frac{1}{5}-\delta}$ immersions of surfaces in 3 dimensions. Rev. Mat. Iberoamericana 34 (2018), 1119-1152. doi: 10.4171/RMI/1019