Balanced metrics on non-Kahler Calabi-Yau threefolds
  We construct balanced metrics on the family of non-K\"ahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on K\"ahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of $k\geq 2$ copies of $S^3\times S^3$. 
Jixiang Fu, Jun Li and Shing-Tung Yau