Irregular $C^{1,\beta}$-surfaces with an analytic metric

We prove that in the class $C^{1,\beta}$ with $\beta<1/13$ it is possible to continuously deform an analytic convex surface of positive Gaussian curvature (or a plane) so as to lose boundedness of the extrinsic curvature in the Pogorelov sense. We demonstrate how to replace the bound $\beta<1/13$ with $\beta<1/7$.