Biharmonic Riemannian submersions from 3-manifolds

TL;DR

This paper extends the characterization of biharmonic maps from surfaces to 3D space forms, showing that biharmonic Riemannian submersions from 3-manifolds are harmonic, generalizing previous results for immersions.

Contribution

It proves that biharmonic Riemannian submersions from 3D space forms to surfaces are equivalent to harmonic ones, generalizing known theorems from immersions to submersions.

Findings

Biharmonic submersions from 3D space forms are harmonic.

Extension of Chen-Ishikawa and Jiang's theorem to submersions.

Applicable to non-positive curvature space forms.

Abstract

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form of non-positive curvature into a surface is biharmonic if and only if it is harmonic.