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

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.