*Published Paper*

**Inserted:** 7 oct 2015

**Last Updated:** 1 dec 2020

**Journal:** Revista matemática Iberoamericana

**Volume:** 34

**Number:** 3

**Pages:** 1119-1152

**Year:** 2018

**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 $\mathbb R^3$ 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.

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