Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form

Abstract

This paper is concerned with the existence theory of isometric immersions of surfaces with negative Gaussian curvature into the three-dimensional Euclidean space. We reformulate the Gauss--Codazzi equations, i.e., the partial differential equations for isometric immersions, into hyperbolic conservation laws for the flows of Chaplygin gas with nonzero source terms. Then, employing theories of invariant regions and compensated compactness, we establish for any given $p \in [2,\infty[$ the existence of $W^{2,p}_{\rm loc}$-isometric immersions of various general families of metrics over infinite strips $\mathbb{R}\times [0,T]$ with arbitrarily large $T$. Such metrics include those of various classical minimal surfaces: helicoid, catenoid, and Enneper surfaces, as well as metrics in isothermal coordinates or of the ``reciprocal-type''. In our fluid dynamical formulation of the isometric immersion problem, we specialise in the case that the two Riemann invariants for the associated hyperbolic conservation law remain bounded and of distinctive signs, and obtain $L^p$-solutions to the initial-boundary value problem via the method of relative entropy with respect to a nonstationary ODE background. The isometric immersions constructed in this paper have second fundamental forms belonging to $L^p_{\rm loc}\setminus L^\infty_{\rm loc}$.