Proper $r$-harmonic functions on the Thurston geometries

TL;DR

This paper constructs explicit proper r-harmonic functions on all five Thurston geometries, advancing understanding of harmonic analysis on these important 3D manifolds.

Contribution

It provides the first explicit constructions of proper r-harmonic functions on all five Thurston geometries, filling a gap in geometric analysis.

Findings

Explicit proper r-harmonic functions on Sol, Nil, SL2, H^2×R, and S^2×R.

New methods for constructing harmonic functions on Thurston geometries.

Enhanced understanding of harmonic analysis in 3D geometric structures.

Abstract

For any positive natural number $r\in\N^+$ we construct new explicit proper $r$-harmonic functions on the celebrated $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.