Consider a smooth connected closed two-dimensional Riemannian manifold $\Sigma$ with positive Gauss curvature. If $u$ is a $C^2$ isometric embedding of $\Sigma$, then $u (\Sigma)$ is convex. In the fifties Nash and Kuiper showed, astonishingly, that this is not necessarily true when the map is $C^1$. It is expected that the threshold at which isometric embeddings "change nature" is the $\frac{1}{2}$-Hoelder continuity of their derivatives, a conjecture which shares a striking similarity with a (recently solved) problem in the theory of fully developed turbulence. In my talk I will review several plausible reasons for the threshold and a very recent work, joint with Dominik Inauen, which indeed shows a suitably weakened form of the conjecture.