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.