  We prove some non-existence theorems for translating solutions to Lagrangian
mean curvature flow. More precisely, we show that translating solutions with an
$L^2$ bound on the mean curvature are planes and that almost-calibrated
translating solutions which are static are also planes. Recent work of D.
Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.
