  We prove uniqueness of smooth isometric immersions within the class of
negatively curved corrugated two-dimensional immersions embedded into
$\mathbb{R}^3$. The main tool we use is the relative entropy method employed in
the setting of differential geometry for the Gauss-Codazzi system. The result
allows us to compare also two solutions to the Gauss-Codazzi system that
correspond to a smooth and a $C^{1,1}$ isometric immersion of not necessarily
the same metric and prove continuous dependence of their second fundamental
forms in terms of the metric and initial data in $L^2$.
