Isometric immersion of completesurface with negative Gauss curvature

The isometric immersion of Riemannian manifold is a fundamental problem in dierential geometry. When the manifold is two dimension and its Gauss curvature is negative, the isometric immersion problem is considered through the Gauss-Codazzi system. It is shown that if the Gauss curvature satises an integrable condition, then there exists a global smooth solution to the Gauss-Codazzi system when the initial data is small. This means the surface has a global isometric immersion in $\mathbb{R}^3$ even the Gauss curvature decays very slowly at infinity.