GEOMETRY OF LAPLACE OPERATOR OR ITS VARIATION TYPE OPERATOR ON RIEMANNIAN MANIFOLD (2003)

我们试图研究以下关于有限型子流形或双调和子流形的主题或问题：(i)在由周期函数z=f(u)生成的x (u, v)=(u cos v, u sin v, f(u))的旋转曲面族中是否存在具有给定平均曲率H的有限型曲面？（ii）是否存在由函数z=f(u)生成的x(u, v)=(u cos v, u sin v, f(u))的旋转曲面族中具有给定高斯曲率K的有限型曲面？（iii）曲面族中是否存在平均曲率恒定的有限型曲面？（iv）是否存在有限型的Willmore曲面？（v-1）空间中具有D ' Atri度量的有限型子流形。（v-2）空间中的双调和子流形（含D ' Atri矩阵）我们将继续研究陈邦彦教授的以下尚未公开的猜想1。确定3维欧几里得空间中所有有限型曲面（Chen猜想1:3维欧几里得空间中仅有的有限型曲面是最小曲面、球面和右圆柱体）。求出n维欧几里德空间中所有的双调和子流形（Chen猜想2：在n维欧几里德空间中只有n（n>3）是调和子流形）。确定4维Minkowsky空间中所有的双调和子流形。

We tried to investigate the following themes or questions with concernig of the finite type submanifolds or biharmonic submanifolds :(i)Are there the finite type surfaces with given mean curvature H in the family of surfaces of revolution x.(u, v)=(u cos v, u sin v, f(u)) generated by the periodic function z=f(u) ?(ii)Are there the finite type surfaces with given Gauss curvature K in the family of surfaces of revolution x(u, v)=(u cos v, u sin v, f(u)) generated by the function z=f(u) ?(iii)Are there the finite type surfaces with constant mean curvature in the family of surfaces ?(iv)Are there the Willmore surfaces of finite type ?(v-1)The finite type submanifolds in the space with D` Atri metric.(v-2)The biharmonic submanifolds in the space with D` Atri metricWe will continue to study in future the following still open conjectures of Prof. Bang-yen Chen1.To determine all of the finite type surfaces in Eucldean space of dimension 3(Chen conjectur 1 : The only finite type surfaces in Eucldean space of dimension 3 are the minimal suefaces, spheres and right cylinders.)2.To determine all of the biharmonic submanifolds in Eucldean space of dimension n(Chen conjectur 2 : The only in Eucldean space of dimension n (n>3) are the harmonc ones.)3.To determine all of the biharmonic submanifolds in Minkowsky space of dimension 4.