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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 .

Polyharmonic, or r-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper r-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion i : S n−1 (R) ֒→ S n is a proper r-harmonic submanifold of S n if and only if the radius R is equal to 1/ √ r. We shall also prove the existence of proper r-harmonic generalized Clifford's tori into the sphere.

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