Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on Bernstein property of a fourth order complex partial
  differential equations   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
$F(\max_{B(R)}\det u_{i\bar j})=o(R).$

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The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg