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

TL;DR

This paper proves that under certain geometric and functional conditions, solutions to a specific fourth-order complex PDE exhibit a Bernstein property, meaning the determinant of the complex Hessian remains constant.

Contribution

It establishes a Bernstein-type result for a class of complex PDEs involving the determinant of the complex Hessian under geometric completeness and curvature bounds.

Findings

Solutions have constant determinant under specified conditions.

The PDE exhibits Bernstein property for complete Kähler metrics.

Conditions on the function F ensure the constancy of the determinant.

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