Surfaces with parallel mean curvature in ℂℙⁿ×ℝ and ℂℍⁿ×ℝ

Abstract

We consider surfaces with parallel mean curvature vector (pmc surfaces) in C P n × R \mathbb {C}P^n\times \mathbb {R} and C H n × R \mathbb {C}H^n\times \mathbb {R} , and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic differential on such surfaces. This is then used in order to show that the anti-invariant pmc 2 2 -spheres of a 5 5 -dimensional non-flat cosymplectic space form of product type are actually the embedded rotational spheres S H 2 ⊂ M ¯ 2 × R S_H^2\subset \bar M^2\times \mathbb {R} of Hsiang and Pedrosa, where M ¯ 2 \bar M^2 is a complete simply-connected surface with constant curvature. When the ambient space is a cosymplectic space form of product type and its dimension is greater than 5 5 , we prove that an immersed non-minimal non-pseudo-umbilical anti-invariant 2 2 -sphere lies in a product space M ¯ 4 × R \bar M^4\times \mathbb {R} , where M ¯ 4 \bar M^4 is a space form. We also provide a reduction of codimension theorem for the pmc surfaces of a non-flat cosymplectic space form.