Isometric Immersion of Surface with Negative Gauss Curvature and the Lax-Friedrichs Scheme

TL;DR

This paper develops a method to construct large $L^ abla$ solutions for the Gauss-Codazzi equations describing isometric immersions of negatively curved surfaces, using a Lax-Friedrichs scheme and compensated compactness.

Contribution

It introduces a novel approach combining finite-difference schemes and compensated compactness to solve the Gauss-Codazzi equations for complex surfaces.

Findings

Established uniform estimates for approximate solutions.

Proved existence of large $L^ abla$ solutions for general surfaces.

Extended previous results to more general surface classes.

Abstract

The isometric immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large $L^\infty$ solution is obtained which leads to a $C^{1,1}$ isometric immersion. The approximate solutions are constructed by the the Lax-Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The $H^{-1}$ compactness is also derived. A compensated compactness framework is applied to obtain the existence of large $L^\infty$ solution to the Gauss-Codazzi equations for the surfaces more general than those in literature.