Harmonicity of the Atiyah-Hitchin-Singer and Eells-Salamon almost   complex structures
  In this paper we describe the oriented Riemannian four-manifolds $M$ for which the Atiyah-Hitchin-Singer or Eells-Salamon almost complex structure on the twistor space ${\mathcal Z}$ of $M$ determines a harmonic map from ${\mathcal Z}$ into its twistor space. 
Johann Davidov, Oleg Mushkarov
math.DG