Annals of PDE

, 4:13 | Cite as

The Anomaly Flow and the Fu-Yau Equation

  • Duong H. Phong
  • Sebastien PicardEmail author
  • Xiangwen Zhang


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.


Hull-Strominger systems Goldstein-Prokushkin fibration Slope parameter \(\alpha '\) Conformally balanced Torsion constraints Curvature Moser iteration \(C^k\) Exponential convergence 



The authors would like to thank the referee for a very careful reading of their paper, and for pointing out the very simple proof of \(C^{2,\alpha }\) estimates.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of CaliforniaIrvineUSA

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