Global Nash-Kuiper theorem for compact manifolds

We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,\theta}$ isometric immersions of compact manifolds with optimal H\"older exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $\theta<1/5$. This extends previous results on embedding 2-discs as well as higher dimensional analogues.