Corrugated Versus Smooth Uniqueness and Stability of Negatively Curved Isometric Immersions

Cleopatra Christoforou

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arXiv · 2303.00359

Corrugated Versus Smooth Uniqueness and Stability of Negatively Curved Isometric Immersions

Cleopatra Christoforou

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Abstract

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 $L^2$.

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Cleopatra Christoforou. 2023-03-01. Corrugated Versus Smooth Uniqueness and Stability of Negatively Curved Isometric Immersions. https://arxiv.org/abs/2303.00359

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