对给定Kahler类中极值度量的研究是复几何中十分重要的研究分支之一，不仅其本身是很基本的问题，重要的是对它的研究会涉及到许多高阶Monge-Ampère型方程,此类方程的研究难度大，理论还很不成熟，需要发展新的手段和方法。本项目拟在前期工作的基础上开展以下方面的研究： 1．具有一定对称性的非Toric流形上稳定性的研究，寻找极值度量的弱解存在的必要条件，2. 高维Abreu方程的正则性和Bernstein性质，在曲率有界的条件下，对 中 S=0的Abreu方程，证明它的解具有Bernstein性质；在数量曲率有界的条件下，证明截口内，最低点邻域的正则性。

One of the central problem in complex geometry is to find certain canonical metrics within a given Kahler class. As examples, the extremal metrics. Not only its own is a basic problem, it also involve many of the higher order Monge-Ampère type equations. Since such equations is very difficult and the theory is still immature, we need develop new means and methods. On the basis of previous works, the project focus on the following aspects:. 1 stability on a certain symmetry, non Toric manifold, a necessary condition for the existence of weak solutions for extremal metrics. 2 Regularity and Bernstein properties of Abreu equation in higher dimension. Under the curvature bounded, prove the Bernstein properties for Abreu equations S=0; for a section, obtain the regularity in a neighbor of the lowest points of the graph.