On positive quaternionic Kahler manifolds with certain symmetry rank

Let M be a positive quaternionic Kahler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HP (m) or Gr2(C (m+2)). This is sharp and optimal, and will complete the classification result of positive quaternionic Kahler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kahler manifolds with a group action and the induction arguments on the dimension of the manifold.