BV weak solutions to Gauss–Codazzi system for isometric immersions

Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature

“…Recently, Chen-Slemrod-Wang in [8] developed a general method, which combines a fluid dynamic formulation of conservation laws for the Gauss-Codazzi system with a compensated compactness framework, to realize the isometric immersions in R 3 with negative Gauss curvature. Christoforou in [11] obtained the small BV solution to the Gauss-Codazzi system with the same catenoid type metric as in [8]. See [18,19,20,28,31,44,46,50] for other related results on surface embeddings.…”

On the decay rate of the Gauss curvature for isometric immersions

“…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.…”

“…It should be mentioned that for discontinuous data of bounded variation, isometric immersions have been established using a different method in [3], the so-called random choice method of Glimm [10], but again with decay rate at least as t −4 . Both methods, the compensated compactness and the random choice, were introduced and developed in the context of the theory of hyperbolic balance laws.…”

“…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.…”