Isometric Immersions and Compensated Compactness

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 ${\mathbb{R}^3}$. 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 ${\mathbb{R}^3}$. 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 ${\mathbb{R}^3}$.

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 ${\mathbb{R}^3}$ satisfying our prescribed initial conditions. To achieve this, we introduce a vanishing viscosity method depending on the features of initial value problems for isometric immersions and present a technique to make the a priori estimates including the L ^∞ control and H ⁻¹–compactness for the viscous approximate solutions. This yields the weak convergence of the vanishing viscosity approximate solutions and the weak continuity of the Gauss-Codazzi system for the approximate solutions, hence the existence of an isometric immersion of the manifold into ${\mathbb{R}^3}$ satisfying our initial conditions. The theory is applied to a specific example of the metric associated with the catenoid.