Some Rigidity Results Related to Monge–Ampère Functions

Jerrard, Robert L.

Canadian journal of mathematics, Tome 62 (2010) no. 2, pp. 320-354

Résumé

The space of Monge–Ampère functions, introduced by J. H. G. Fu, is a space of rather rough functions in which the map $u\,\mapsto \,\text{Det}\,{{\text{D}}^{2}}u$ is well defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge–Ampère functions. We also prove that if a Monge–Ampère function $u$ on a bounded set $\Omega \,\subset \,{{\mathbb{R}}^{2}}$ satisfies the equation $\text{Det}\,{{D}^{2}}u\,=\,0$ in a particular weak sense, then the graph of $u$ is a developable surface, and moreover $u$ enjoys somewhat better regularity properties than an arbitrary Monge–Ampère function of 2 variables.

Détail

DOI : 10.4153/CJM-2010-019-8

Mots-clés : 49Q15, 53C24