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

TL;DR

This paper establishes the existence of global weak solutions for isometric immersions of negatively curved surfaces with two classes of metrics into 3D space using advanced PDE techniques.

Contribution

It introduces a novel approach to solve the Gauss-Codazzi equations for surfaces with negative Gauss curvature via vanishing viscosity and compensated compactness.

Findings

Global weak solutions in $L^ Infty$ for the Gauss-Codazzi equations.

Construction of invariant regions for catenoid and helicoid type metrics.

Establishment of $L^ Infty$ isometric immersions for given metrics.

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.