Nonfattening condition for the generalized evolution by mean curvature and applications

We prove a non fattening condition for a geometric evolution described by the level set approach. This condition is close to those of Soner \cite{soner93} and Barles, Soner and Souganidis \cite{bss93} but we apply it to some unbounded hypersurfaces. It allows us to prove uniqueness for the mean curvature equation for graphs with convex at infinity initial data, without any restriction on its growth at infinity, by seeing the evolution of the graph of a solution as a geometric motion.