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

Camillo De Lellis, Dominik Inauen, László Székelyhidi

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 $…