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

Abstract

The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in L ∞ are obtained through the vanishing viscosity method and the compensated compactness framework. The L ∞ uniform estimate and H −1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in L ∞ to the Gauss-Codazzi equations yield the C 1,1 isometric immersions of surfaces with the given metrics.

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.…”

“…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.…”

“…The advantage of the one given in [4] is its relative simplicity and the precise estimates for (u, v) from above, below (respectively). Next, we show a crucial H −1 loc estimate which is needed to apply the method of compensated compactness following a standard argument say as given in the paper of Cao, Huang and Wang [1] and use the entropy, entropy flux pair…”

“…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.…”