On the Existence of $C^{1,1}$ Isometric Immersions of Several Classes of Negatively Curved Surfaces Into $\mathbb{R}^3$

TL;DR

This paper proves the existence of $C^{1,1}$ isometric immersions for various negatively curved surfaces into $ ^3$, using advanced mathematical techniques to handle the geometric complexities involved.

Contribution

It establishes the existence of such immersions for multiple classes of negatively curved metrics, expanding the understanding of surface embeddings in Euclidean space.

Findings

Existence of $C^{1,1}$ isometric immersions for hyperbolic and generalized metrics.

Application of compensated compactness and hyperbolic conservation laws.

Insights into geometric quantities of negatively curved surfaces.

Abstract

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These include the standard hyperbolic plane, generalised helicoid-type metrics and generalised Enneper metrics. Our proof is based on the method of compensated compactness and invariant regions in hyperbolic conservation laws, together with several observations on the geometric quantities (Gauss curvature, metric components etc.) of negatively curved surfaces.