ABSTRACT

The isometric immersion of Riemannian manifold is a fundamental problem in differential 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 satisfies an integrable condition, then the surface has a global isometric immersion in $R^3$ even the Gauss curvature decays very slowly at infinity.