Biharmonic Functions on the Classical Compact Simple Lie Groups

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 $$\mathbf{SU}(n)$$ , $$\mathbf{SO}(n)$$ and $$\mathbf{Sp}(n)$$ . 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 $${\mathbb S}^3$$ and on the hyperbolic space $${\mathbb H}^3$$ .