On harmonic tori in compact rank one symmetric spaces

In this paper we prove that, in contrast with the SnSn and CPnCPn cases, there are harmonic 2-tori into the quaternionic projective space HPnHPn which are neither of finite type nor of finite uniton number; we also prove that any harmonic 2-torus in a compact Riemannian symmetric space which can be obtained via the twistor construction is of finite type if and only it is constant; in particular, we conclude that any harmonic 2-torus in CPnCPn or SnSn which is simultaneously of finite type and of finite uniton number must be constant.