  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.
