A note on Bernstein property of a fourth order complex partial differential equations

Abstract: 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 $F(\max_{B(R)}\det u_{i\bar j})=o(R).$