Biharmonic functions on the classical compact simple Lie groups

TL;DR

This paper constructs new proper biharmonic functions on classical compact Lie groups and relates them to harmonic morphisms, providing duality interpretations on spheres and hyperbolic spaces.

Contribution

It introduces new families of proper biharmonic functions on Lie groups and develops a duality principle linking these functions to harmonic morphisms.

Findings

New biharmonic functions on $ ext{SU}(n)$, $ ext{SO}(n)$, and $ ext{Sp}(n)$

Duality principle connecting biharmonic functions with harmonic morphisms

Interpretation of examples on spheres and hyperbolic spaces

Abstract

The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups $\SU n$, $\SO n$ and $\Sp n$. We work in a geometric setting which connects our study with the theory of submersive harmonic morphisms. We develop a general duality principle and use this to interpret our new examples on the Euclidean sphere $\s ^3$ and on the hyperbolic space $\H^3$.