Linear Strain Tensors and Optimal Exponential of thickness in Korn's Inequalities for Hyperbolic Shells

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces to obtain $L^2$ regularity solutions. Then the rigidity results on the strain tensor of the middle surface are implied by the $L^2$ regularity for non-characteristic regions. Finally, we obtain the optimal constant in the first Korn inequality scales like $h^{4/3}$ for hyperbolic shells, generalizing the assumption that the middle surface of the shell is given by a single principal system in the literature.