On the existence of \(C^{1,1}\)-isometric immersions of several classes of negatively curved surfaces into \(\mathbb{R}^3\) (Q2303868)

The main goal of this paper is to prove the existence of \(C^{1,1}\) isometric immersions for three families of negatively curved Riemannian surfaces \((M, g)\) into 3-dimensional Euclidean space: \begin{itemize} \item[(1)] the standard hyperbolic plane; \item[(2)] helicoid-type metrics of the form \(g = (A y^2 + B y + C)\, dx^2 + dy^2\) where \(A>0\), \(B \leq 0\), and \(B^2 - 4AC < 0\); \item[(3)] generalized Enneper metrics of the form \(g = (1 + x^2 + y^2)^{\alpha} (dx^2 + dy^2)\) for \(\alpha \in [1, 10]\) . \end{itemize} The primary tools used to prove these results are the theory of compensated compactness and the method of invariant regions in hyperbolic conservation laws. It is also shown that this method fails due to the non-existence of invariant regions for certain reciprocal-type metrics of the form \(g = E(y)\, dx^2 + E(y)^{-1}\, dy^2\). It should be noted that the \(C^{1,1}\) isometric immersions established here are only ``nearly global''. For instance, the immersions constructed for the upper half-plane model for the hyperbolic plane are valid in any half-space of the form \[ \{(x, y): y > y_0 \} \] for any \(y_0 > 0\).