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

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.

25 pages + 3 figures. Supplementary materials on Wolfram Cloud

References in corpus (2)

• Global Weak Rigidity of the Gauss-Codazzi-Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity
• Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature