Isometric Immersions via Compensated Compactness for Slowly Decaying Negative Gauss Curvature and Rough Data

Abstract

In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-\delta/2}$ where $\delta\in(0,4)$ suffices.