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

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