# Corrugated Versus Smooth Uniqueness and Stability of Negatively Curved Isometric Immersions

## TL;DR

This paper establishes the uniqueness and stability of negatively curved isometric immersions into three-dimensional space, using the relative entropy method to compare smooth and less regular solutions of the Gauss-Codazzi system.

## Contribution

It introduces a novel application of the relative entropy method to prove uniqueness and continuous dependence for negatively curved isometric immersions in differential geometry.

## Key findings

- Uniqueness of smooth isometric immersions with negative curvature.
- Continuous dependence of second fundamental forms on the metric and initial data.
- Comparison of solutions with different regularity levels for the Gauss-Codazzi system.

## 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 $L^2$.