利用椭圆偏微分方程理论来解决流形上的若干存在性和唯一性问题，是微分几何和复几何中的重要问题。我们希望讨论若干这类问题：(1)研究从双球面到球面嵌入超曲面的预给定 Gauss-Kronecker 曲率，平均曲率和第 m 个平均曲率问题；(2) 讨论形式型Calabi-Yau方程以及non-Kahler几何中典则度量的存在性问题；(3)讨论广义Gauduchon度量存在性方程以及其在复流形上的应用；(4)考虑高余维平均曲率流的自收缩解的 Bernstein 型定理。

It is a very important method that we can solve the existence and uniqueness problems in differential geometry and complex geometry, using the elliptic partial differential equation theory. For example, Minkowski problem, Calabi-Yau theorem, the existence of K¨ahler-Einstein metric, Yamabe problem, minimal manifold problem, Ricci flow and Poincar′e conjuncture, mean-curvature flow and Bernstein type theorems, are all important issues of this aspect. In our plan, we will discuss several problems of this kind. 1. Research the embeddings from the product of two unit spheres to some high dimensional sphere with prescribed Gauss-Kronecker curvature, mean curvature or the m-th mean curvatures. 2. Discuss the form-type Calabi-Yau equations and the existence of the canonical metrics in non-Kahler geometry. 3. Discuss the Generalised Gauduchon metrics on Hermitian manifolds and its applications to complex manifolds. 4. Consider the Bernstein type theorems of the self-shrinking solutions corresponding to the mean curvature flow in high co-dimension space.