Abstract
Let \({f:\Omega \rightarrow \mathbb{R}}\) be a smooth function on a domain \({\Omega \subset \mathbb{C}^n}\) with its Hessian matrix \({\left( \frac{\partial^2 f}{\partial z^i \partial\bar{z}^j}\right)}\) positive Hermitian. In this paper, we investigate a class of partial differential equations
where Δ and \({\| \cdot \|}\) are the Laplacian and tensor norm, respectively, with respect to the metric \({G = \sum f_{i\bar{j}} \,dz^i \otimes d\bar{z}^j}\), and β > 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 \({\det (f_{i\bar{j}})}\) and the Ricci curvature are bounded.
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An erratum to this article can be found at http://dx.doi.org/10.1007/s00025-010-0046-0
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Chen, G., Sheng, L. A Bernstein Property of a Class of Fourth Order Complex Partial Differential Equations. Results. Math. 58, 81–92 (2010). https://doi.org/10.1007/s00025-010-0030-8
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DOI: https://doi.org/10.1007/s00025-010-0030-8