The Anomaly flow and the Fu-Yau equation
  The Anomaly flow is shown to converge on toric fibrations with the Fu-Yau ansatz, for both positive and negative values of the slope parameter $\alpha'$. This implies both results of Fu and Yau on the existence of solutions for Hull-Strominger systems, which they proved using different methods depending on the sign of $\alpha'$. It is also the first case where the Anomaly flow can even be shown to exist for all time. This is in itself remarkable from the point of view of the theory of fully nonlinear partial differential equations, as the elliptic terms in the flow are not concave. 
Duong H. Phong, Sebastien Picard, Xiangwen Zhang
math.DG math.AP math.CV