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

Abstract

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 $L^{\infty }$ are obtained through the vanishing viscosity method and the compensated compactness framework. The $L^{\infty }$ 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 $L^{\infty }$ to the Gauss-Codazzi equations yield the $L^{\infty }$ isometric immersions of surfaces with the given metrics.

24 pages, 5 figures,

Cited by in corpus (5)

• Isometric Immersions via Compensated Compactness for Slowly Decaying Negative Gauss Curvature and Rough Data
• On the Existence of $C^{1,1}$ Isometric Immersions of Several Classes of Negatively Curved Surfaces Into ${\mathbb{R}}^3$
• Weak continuity of curvature for connections in $L^p$
• Isometric Immersion of Complete Surfaces with Slowly Decaying Negative Gauss Curvature
• Isometric Immersion of Surface with Negative Gauss Curvature and the Lax-Friedrichs Scheme