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

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.