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

TL;DR

This paper extends the application of compensated compactness to isometric immersions of 2D manifolds with negative Gauss curvature, showing that slower decay rates than previously required are sufficient for the method to work.

Contribution

It demonstrates that the decay rate of Gauss curvature can be relaxed from $t^{-4}$ to $t^{-2-rac{eta}{2}}$, broadening the applicability of the method.

Findings

Slower decay rates of Gauss curvature are sufficient for isometric immersion.

The method applies to rough data with less restrictive decay conditions.

Extension of previous results to a wider class of manifolds.

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.