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 ${\mathbf{R}}_{+}^{\ast }$, 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 Kähler metric $g=\partial \bar{\partial }u$. We show that the above PDE's has a Bernstein property, i.e $\det (u_{i\bar{j}})=constant$ on $\Omega$, provided that $g$ is complete, the Ricci curvature of $g$ is bounded below and $F$ satisfies ${\inf }_{t\in {\mathbf{R}}^{+}}(2t{F}^{{}^{\prime }}(t)+\frac{F(t{)}^2}{n})>\frac{1}{4}$ and $F({\max }_{B(R)}\det u_{i\bar{j}})=o(R).$