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

黎曼流形上拉普拉斯算子或其变分型算子的几何结构(2003)

基本信息

批准号：
12640078
负责人：
ISHIKAWA Susumu
金额：
$ 2.37万
依托单位：
SAGA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

项目摘要

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.

本文研究了有限型子流形或双调和子流形的下列问题：（i）在旋转曲面族x中是否存在具有给定平均曲率H的有限型曲面.（u，v）=（u cos v，u sin v，f（u））由周期函数z=f（u）生成？（ii）在由函数z=f（u）生成的旋转曲面族x（u，v）=（u cos v，u sin v，f（u））中，是否存在具有给定Gauss曲率K的有限型曲面？（iii）曲面族中是否存在具有常平均曲率的有限型曲面？（iv）是否存在有限型Willmore曲面？（v-1）D` Atri度量空间中的有限型子流形.（v-2）D` Atri度量空间中的双调和子流形我们将继续研究陈邦彦教授的下列尚未解决的问题：1.确定3维Eucldean空间中的所有有限型曲面（陈邦彦教授的结论1：3维Eucldean空间中的唯一有限型曲面是极小曲面、球面和直柱面）。2.确定了n维欧氏空间中的所有双调和子流形（陈章2：n维欧氏空间（n>3）中只有调和子流形）。3.确定了四维Minkowski空间中的所有双调和子流形。