Isometric Immersions and Compensated Compactness

TL;DR

This paper introduces a novel approach combining fluid dynamics and compensated compactness to solve the isometric immersion problem for negatively curved surfaces in three-dimensional space.

Contribution

It develops a new framework that ensures weak continuity of the Gauss-Codazzi system, enabling the realization of 2D Riemannian manifolds as surfaces in R^3.

Findings

Existence of a C^{1,1} isometric immersion for negatively curved surfaces.

A general method for initial/boundary value problems in isometric immersions.

Application of compensated compactness to differential geometry problems.

Abstract

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T