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. T

25 pages, 6 figues

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• On the Existence of $C^{1,1}$ Isometric Immersions of Several Classes of Negatively Curved Surfaces Into ${\mathbb{R}}^3$
• Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity
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