PMC Biconservative Surfaces in Complex Space Forms

TL;DR

This paper investigates PMC surfaces in complex space forms, establishing conditions for biconservativity, deriving a Simons type formula, and providing classification and construction results for such surfaces and submanifolds.

Contribution

It characterizes PMC surfaces as biconservative if and only if totally real in non-flat complex space forms and introduces new formulas and classification results.

Findings

PMC surfaces in non-flat complex space forms are biconservative iff totally real.

A Simons type formula for a vector field from the mean curvature vector is derived.

Rigidity and reduction codimension results for CMC and PMC biconservative surfaces are proved.

Abstract

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.