# Biharmonic constant mean curvature surfaces in Killing submersions

## TL;DR

This paper characterizes proper biharmonic constant mean curvature surfaces within Killing submersions and classifies proper biharmonic Hopf cylinders, advancing understanding of biharmonic geometry in these manifolds.

## Contribution

It provides a new characterization of proper biharmonic CMC surfaces and classifies proper biharmonic Hopf cylinders in Killing submersions.

## Key findings

- Characterization of proper biharmonic CMC surfaces in Killing submersions
- Classification of proper biharmonic Hopf cylinders
- Enhanced understanding of biharmonic surfaces in Riemannian submersions

## Abstract

A $3$-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part, we also classify the proper biharmonic Hopf cylinders in a Killing submersion.