A note on Biharmonic functions on the Thurston geometries

TL;DR

This paper constructs explicit proper biharmonic functions on various 3D Thurston geometries, expanding the understanding of biharmonic functions in these complex geometric contexts.

Contribution

It introduces new explicit proper biharmonic functions on five different Thurston geometries, which were not previously known.

Findings

Explicit proper biharmonic functions on $ ext{Sol}$, $ ext{Nil}$, $ ext{SL}_2$, $H^2 imes n$, and $S^2 imes n$.

Advances the understanding of biharmonic functions in Thurston geometries.

Provides tools for further geometric analysis in these spaces.

Abstract

We construct new explicit proper biharmonic functions on the $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.