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 is ometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails up to exponent $\theta<1 /5$. this extends previous results on embedding 2-discs as well as higher dimensional analogues. this is a joint work with prof. laszlo szekelyhidi in university of leipzig.

Wentao Cao, graduated from AMSS, Chinese Academy of Sciences, now is the post-doc at Universität Leipzig, joint work with Prof. Laszlo Szekelyhidi. His research mainly focuses on isometric imbedding problem and nonlinear hyperbolic system of conservation laws. He was rewarded the Excellent Ph.D. thesis in Chinese Academic Sciences in 2017, Dean Special Scholarship of Chinese Academy of Sciences in 2016 and so on.