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We prove that, given a $C^2$ Riemannian metric $g$ on the $2$-dimensionaldisk $D_2$, any short $C^1$ immersion of $(D_2,g)$ into $\mathbb R^3$ can beuniformly approximated with $C^{1,\alpha}$ isometric immersions for any $\alpha< \frac{1}{5}$. This statement improves previous results by Yu.F. Borisov andof a joint paper of the first and third author with S. Conti.

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