仿射微分几何是微分几何的一个重要分支。完备仿射超曲面是仿射微分几何的重要组成部分，涉及到很多深刻的非线性偏微分方程，历来受到几何学家与偏微分方程学家的重视。由于所涉及问题的研究难度较大，目前理论还很不成熟，所以急需发展新的理论和方法。本项目拟利用仿射技巧研究完备仿射超曲面的Bernstein问题，以及一些相关的平均曲率流的完备孤立子(soliton）的分类问题。具体研究内容如下：1．alpha相对超曲面的Bernstein问题，其中包括仿射极大超曲面、仿射Kähler极小曲面的Bernstein问题等；2.伪欧氏空间中平均曲率流的类空entire translating soliton的分类问题；3．仿射平均曲率流的完备soliton的构造以及分类问题。本项目将发展一些新的方法和技巧，进而丰富和发展仿射微分几何、子流形几何和Monge-Ampère方程的理论。

Affine differential geometry is one of important branches of differential geometry. Complete affine hypersurfaces is an important part of affine differential geometry. Because it involves deep theories of nonlinear partial differential equations, the specialists in PDEs and the geometers have always concerned with them. As such problems are very difficult, known knowledge on this aspect is limited up to date. The current theory is still immature, so it is urgent that we need to develop some new theories and methods. The project intends to use affine geometric methods to study the Bernstein problems of complete affine hypersurfaces, and the classification of complete solitons to the related mean curvature flows. Our specific research contents are as follows:.1.Bernstein problems of alpha relative hypersurfaces, eg. affine maximal hypersurfaces and alpha relative affine extremal hypersurfaces;.2.Classification of spacelike entire translating solitons to the mean curvature flow in pseudo-Euclidean space; .3.Construction and classification of solitons to the affine mean curvature flow. .The project aims to achieve substantive progress on this aspect and develop some new methods and techniques. It will be helpful for us to enrich and develop theories of affine differential geometry, submanifold geometry and Monge-Ampère equation.