Surfaces with parallel mean curvature vector in complex space forms

Details Surfaces with parallel mean curvature vector in complex space forms Surfaces with parallel mean curvature vector in complex space forms

2010-11-25

arxiv.org/abs/1011.5892v1

Mathematics

Differential Geometry

14 pages

Scientific paper

We consider a quadratic form defined on the surfaces with parallel mean curvature vector of an any dimensional complex space form and prove that its $(2,0)$-part is holomorphic. When the complex dimension of the ambient space is equal to $2$ we define a second quadratic form with the same property and then determine those surfaces with parallel mean curvature vector on which the $(2,0)$-parts of both of them vanish. We also provide a reduction of codimension theorem and prove a non-existence result for $2$-spheres with parallel mean curvature vector.

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