Parallel mean curvature biconservative surfaces in complex space forms

Abstract

In this paper, we consider PMC surfaces in complex space forms, and study the interaction between the notions of PMC, totally real and biconservative. We first consider PMC surfaces in a non‐flat complex space form and 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 and use it to prove a rigidity result for CMC biconservative surfaces in two‐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 examples of CMC non‐PMC biconservative submanifolds from the Segre embedding and discuss when they are proper‐biharmonic.