For a smooth strictly plurisubharmonic function $u$ on a open set $\Omega\subset\mathbb{ C}^{ n}$ and $F$ a $C^{ 1}$ nondecreasing function on $\mathbf{ R}^{ *}_{ +}$, we investigate the complex partial differential equations $$\Delta_{ g}\log\det(u_{ i\bar j})=F(\det(u_{ i\bar j}))\Vert\nabla_{ g}\log\det(u_{ i\bar j})\Vert_{ g}^{ 2},$$ where $\Delta_{ g}$, $\Vert . \Vert_{ g}$ and $\nabla_{ g}$ are the Laplacian, tensor norm and the Levi-Civita connexion , respectively, with respect to the K\"ahler metric $g=\partial\bar\partial u$. We show that the above PDE's has a Bernstein property, i.e $\det(u_{ i\bar j})=\hbox{ constant}$ on $\Omega$, provided that $g$ is complete, the Ricci curvature of $g$ is bounded below and $F$ satisfies $\inf_{ t\in\mathbf{ R}^{ +}}(2tF^{ '}(t)+{ F(t)^{ 2}\over n})>{ 1\over 4}$ and $F(\max_{ B(R)}\det u_{ i\bar j})=o(R).$
Categories: math.DG math.AP