Abstract.
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 tend to degenerate along holomorphic curves described by solutions of ODE. The goal of this article is to describe the "Monge-Ampère geometry" of the jet-space that corresponds to this phenonemon. 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 riemaniann lamination.
Article PDF
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Additional information
Submitted: December 1995, revised version: September 1996, final version: March 1997
Rights and permissions
About this article
Cite this article
Labourie, F. Problèmes de Monge-Ampère, courbes holomorphes et laminations. GAFA, Geom. funct. anal. 7, 496–534 (1997). https://doi.org/10.1007/s000390050017
Issue Date:
DOI: https://doi.org/10.1007/s000390050017