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.

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.