Global Nash-Kuiper's theorem to compact manifolds

In this talk, I will present some recent results regarding isometric embedding. I first review some conclusions on $C^{1,\theta}$ isometric immersions and isometric extension and related problem. Then I will show our 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. The presented results are joint work with my mentor Prof. Dr. Szekelyhidi in Leipzig University.