# dg.differential geometry – About the local isometric integration counterpart of Pogorelov-Nadirashvili-Yuan

In Pogorelov's article "An example of a two-dimensional Riemannian metric that does not allow local realization in E3 Dokl Akad Nauk SSSR Tom 198 (1), 42-43 (1971), translation into English in Soviet mathematics Dokl. , 729 -730 (1971) 123 "and the Nadirashvili-Yuan document" Improvement of the isometric inlaid counter-example of Pogorelov, Calc. Var, Partial differential equations 32 (2008), No. 3, 319-323, 53C45 ", a counterexample of local isometric embedding for $$C2,1$$ Riemannian meter As far as I know, those two documents are the only source of such counter-example for local isometric inclusion.

There is a crucial estimate about the second derivatives of the graph: $$min _ {- c leq t_1 leq c} h_ {11} (t_1, b) leq (Mm) frac {b ^ 2} {c ^ 2}$$ (see page 322 of the Nadirashvili-Yuan article). I can not verify this estimate.

Did someone verify the previous counterexample in details or knew how to obtain the previous estimate?