A Bernstein Property of a Class of Fourth Order Complex Partial Differential Equations

作者：Chen Gang^*; Sheng Li

来源：Results in Mathematics, 2010, 58(1-2): 81-92.

DOI：10.1007/s00025-010-0030-8

• Complex PDE of fourth order
• Bernstein property
• complete Kahler manifolds

Let f : Omega -> R be a smooth function on a domain Omega subset of C(n) with its Hessian matrix (partial derivative(2)f/partial derivative z(i) partial derivative Z(j)) positive Hermitian. In this paper, we investigate a class of partial differential equations
Delta ln det(f(ij)(-)) = beta parallel to grad ln det(f(ij)(-))parallel to(2),
where Delta and parallel to.parallel to are the Laplacian and tensor norm, respectively, with respect to the metric G = Sigma fij dz(i) circle times d(z)(-j), and beta > 1 is some real constant depending on the dimension n. We prove that the above PDEs have a Bernstein property when the metric G is complete, provided that det(f(ij)(-) and the Ricci curvature are bounded.

• 出版日期2010-9
• 单位四川大学