On non-K\"ahler compact complex manifolds with balanced and   astheno-K\"ahler metrics
  In this note we present, for every $n \geq 4$, a non-K\"ahler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-K\"ahler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$. 
Adela Latorre, Luis Ugarte
math.DG