Corrugated versus smooth uniqueness and stability of negatively curved isometric immersions

We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into R 3 \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 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 L^2 .