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Surfaces with parallel mean curvature vector in complex space forms. (English) Zbl 1272.53051

For a surface immersed in a complex space form with complex structure \(J\) and constant holomorphic sectional curvature \(\rho\), the author introduces a quadratic form defined by \[ Q(X,Y)=8|H|^2\langle\sigma (X,Y),H\rangle+3\rho\langle JX, H\rangle\langle JY, H\rangle, \] where \(\sigma\) denotes the second fundamental form and \(H\) the mean curvature vector of the surface. The paper gives a good study of such surfaces by using this quadratic form and another one when the target space has complex dimension \(2\).
The interesting results obtained in the paper include: (1) The \((2, 0)\)-part of the quadratic form of a surface in a complex space form with parallel mean curvature vector field is holomorphic; (2) For an immersed surface in \(\mathbb{C}^n\) with parallel mean curvature vector field the quadratic form has zero \((2, 0)\)-part if and only if the surface is pseudo-umbilical; (3) A surface in a complex space form of nonzero constant holomorphic sectional curvature with nonzero parallel mean curvature vector field is either totally real and pseudo-umbilical or it is not pseudo-umbilical and lies in a complex space form of complex dimension no more than \(5\).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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