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The isometric immersion of two-dimensional Riemannian manifolds or surfacesin the three-dimensional Euclidean space is a fundamental problem indifferential geometry. When the Gauss curvature is negative, the isometricimmersion problem is considered in this paper through the Gauss-Codazzi systemfor the second fundamental forms. It is shown that if the Gauss curvaturesatisfies an integrability condition, the surface has a global smooth isometricimmersion in the three-dimensional Euclidean space even if the Gauss curvaturedecays very slowly at infinity. The new idea of the proof is based on the novelobservations on the decay properties of the Riemann invariants of theGauss-Codazzi system. The weighted Riemann invariants are introduced and acomparison principle is applied with properly chosen control functions.

Isometric Immersion of Complete Surfaces with Slowly Decaying Negative Gauss Curvature