Characterizing bi-harmonic homomorphisms in three-dimensional unimodular Lie groups

This study is dedicated to the classification of bi-harmonic homomorphisms $\varphi\colon(G,g)\to (H,h)$, where $G$ and $H$ represent connected and simply connected three-dimensional unimodular Lie groups, while $g$ and $h$ denote left invariant Riemannian metrics.