Isometric Immersions and Compensated Compactness

Abstract

Abstract. A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold 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 elliptichyperbolic 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. 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 apriori 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 R 3 satisfying our initial conditions.

On the decay rate of the Gauss curvature for isometric immersions

“…Having these, we observe that the compensated compactness framework as described in Theorem 4.1 of [2] is applicable. This implies that there exists a subsequence, still labeled (l μ , m μ , n μ ) that converges weak* in L ∞ ( ) to (l,m,ñ) as μ → 0 and the limit (l,m,ñ) is a bounded weak solution of the Gauss-Codazzi system in the domain .…”

“…This means that the immersion is smooth enough so that the Gauss curvature is well defined. The main difference in these results [2,1,3,4] is the rate of the Gauss curvature considered in each work and as it is mentioned later the case of the slower decay rate t −(2+δ) of Hong [13] is the one promoted here.…”

“…In two papers Chen, Slemrod and Wang [2] and Cao, Huang and Wang [1] have used the method of compensated compactness to establish global isometric immersions into R 3 for two dimensional Riemannian manifolds for rough data.…”

“…This article serves as a survey of recent applications [2,1,3,4] of the method of compensated compactness to prove the global isometric immersion of a two dimensional Riemannian manifold with slowly decaying negative Gauss curvature into three dimensional Euclidean space. The immersion will lie in the Sobolev space W 2,∞ loc and hence will be locally in C 1,1 , cf.…”

“…Now, the question raised is whether one can capture such a result in the setting of non-smooth immersions. A search for "corrugated immersions" that ask the data to be "rough" and not in C 1 is the issue pursued in the recent articles [2,1,3,4]. In particular, we focus our discussion on data in L ∞ for which the method of compensated compactness has been successfully applied.…”