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

For a smooth strictly plurisubharmonic function u on an open set Omega subset of C-n and F a C-1 nondecreasing function on R-+(*), we investigate the complex partial differential equations Delta(g) log det (u(i (j) over bar)) = F(det(u(i (j) over bar))) parallel to del(g) log det(u(i (j) over bar))parallel to(2)(g), where Delta(g), parallel to.parallel to(g) and del(g) are the Laplacian, tensor norm and the Levi-Civita connexion, respectively, with respect to the Uhler metric g = partial derivative(partial derivative) over baru. We show that the above PDE's has a Bernstein property, i.e. det(u(i (j) over bar)) is constant on Omega, provided that g is complete, the Ricci curvature of g is bounded below and F satisfies inf(t is an element of R)+ (2tF' (t) + F(t)(2)/n) > 1/4 and F(max(R(R))detu(i (j) over bar)) = o(R). (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.