Abstract view
Some Rigidity Results Related to Monge—Ampère Functions


Published:20091204
Printed: Apr 2010
Abstract
The space of MongeAmpère functions, introduced by J. H. G. Fu, is
a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2
u$ 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 MongeAmpère functions. We also
prove that if a MongeAmpère function $u$ on a bounded set
$\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ 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 MongeAmpère function of $2$ variables.