Surfaces with parallel mean curvature vector in complex space forms

TL;DR

This paper investigates surfaces with parallel mean curvature vectors in complex space forms, introducing quadratic forms that are holomorphic and characterizing special surfaces, including a non-existence result for certain spheres.

Contribution

It introduces a new quadratic form on such surfaces, proves its holomorphicity, and characterizes surfaces with vanishing parts of these forms, including a non-existence theorem for 2-spheres.

Findings

The (2,0)-part of the quadratic form is holomorphic.

Characterization of surfaces with vanishing (2,0)-parts of the quadratic forms.

Non-existence of 2-spheres with parallel mean curvature vector.

Abstract

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.