  In this paper, we study the smooth isometric immersion of a complete simply
connected surface with a negative Gauss curvature in the three-dimensional
Euclidean space. For a surface with a finite total Gauss curvature and
appropriate oscillations of the Gauss curvature, we prove the global existence
of a smooth solution to the Gauss-Codazzi system and thus establish a global
smooth isometric immersion of the surface into the three-dimensional Euclidean
space. Based on a crucial observation that some linear combinations of the
Riemann invariants decay faster than others, we reformulate the Gauss-Codazzi
system as a symmetric hyperbolic system with a partial damping. Such a damping
effect and an energy approach permit us to derive global decay estimates and
meanwhile control the non-integrable coefficients of nonlinear terms.
