Summary: Riemann’s Uniformization theorem is a classical tool for the study of elliptic problems on surfaces. Usually, the use of this theorem reflects the fact that the situation can be translated in a pseudo-holomorphic language: the solutions of the problem appearing as holomorphic curves for a suitable almost complex structure in a jet space. Often, the lack of compactness of the space of solutions of bounded energy is remarkably described by Gromov’s compactness theorem on holomorphic curves. On the other hand for other problems, usually related to Monge-Ampère equations, a different type of lack of compactness appears; solutions with bounded energy converge and, furthermore, it is possible to describe what happens when the energy goes to infinity: the solutions of ODE.
The goal of this article is to describe the “Monge-Ampère geometry” of the jet-space that corresponds to this phenomenon. We prove compactness results for the solutions of these problems, and show examples and applications of our technique. Furthermore, a moduli space of pointed solutions is exhibited with its structure of a riemannian lamination.