  In this article we consider PMC surfaces in complex space forms, and we study
the interaction between the notions of PMC, totally real and biconservative. We
first consider PMC surfaces in non-flat complex space forms and we prove that
they are biconservative if and only if totally real. Then, we find a Simons
type formula for a well-chosen vector field constructed from the mean curvature
vector field. Next, we prove a rigidity result for CMC biconservative surfaces
in 2-dimensional complex space forms. We prove then a reduction codimension
result for PMC biconservative surfaces in non-flat complex space forms. We
conclude by constructing from the Segre embedding examples of CMC non-PMC
biconservative submanifolds, and we also discuss when they are
proper-biharmonic.
