Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature

The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in are obtained through the vanishing viscosity method and the compensated compactness framework. The uniform estimate and H (-1) compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in to the Gauss-Codazzi equations yield the C (1,1) isometric immersions of surfaces with the given metrics.