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

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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.