A Nash-Kuiper theorem for \(C^{1,\frac{1}{5}-\delta}\) immersions of surfaces in 3 dimensions (Q1989542)

It is well known that any short $C^1$ immersion of $n$-manifolds into $\mathbb{R}^{n+1}$ can be uniformly approximated by a $C^1$ isometric immersion. Since the same result does not hold for $C^2$ embeddings, it is an open question for which $\beta\in(0,1)$ there can be found a $C^{1,\beta}$ embedding. The main goal of the presented paper is to show that for the case of $n=2$, $\beta$ can be equal to at least $1/5$, which improves the result of \textit{Yu. F. Borisov} [Sib. Mat. Zh. 45, No. 1, 25--61 (2004; Zbl 1054.53081); translation in Sib. Math. J. 45, No. 1, 19--52 (2004)].