Estimates for a geometric flow for the Type IIB string


It is shown that bounds of all orders of derivative would follow from uniform bounds for the metric and the torsion 1-form, for a flow in non-Kähler geometry which can be interpreted as either a flow for the Type IIB string or the Anomaly flow with source term and zero slope parameter. A key ingredient in the proof is a formulation of this flow unifying it with the Ricci flow, which was recently found.

This is a preview of subscription content, access via your institution.


  1. 1.

    Bedulli, L., Vezzoni, L.: A parabolic flow of balanced metrics. J. Reine Angew. Math. 723, 79–99 (2017)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bedulli, L., Vezzoni, L.: Stability of geometric flows of closed forms. Adv. Math. 364, 107030, 29 (2020)

  3. 3.

    Bryant, R., Xu, F.: Laplacian flow for closed \(G_2\) structures: Short-time Behavior. arXiv:1101.2004

  4. 4.

    Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nuclear Phys. B 258(1), 46–74 (1985)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Dinew, S., Kolodziej, S.: Liouville and Calabi-Yau type theorems for complex Hessian equations. Amer. J. Math. 139(2), 403–415 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Fei, T., Huang, Z., Picard, S.: The Anomaly flow over Riemann surfaces. arXiv:1703.10067. Int. Math. Res. Not. rnz076

  7. 7.

    Fei, T., Picard, S.: Anomaly flow and T-duality. arXiv:1903.08768, to appear in Pure Appl. Math. Q

  8. 8.

    Fei, T., Phong, D.H.: Unification of the Kähler-Ricci and Anomaly flows. Differential Geometry, Calabi-Yau Theory, and General Relativity, 89-104, Surv. Differ. Geom., 22, International Press (2018)

  9. 9.

    Fei, T., Yau, S.T.: Invariant solutions to the Strominger system on complex Lie groups and their quotients. Comm. Math. Phys. 338(3), 1183–1195 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fu, J.X., Yau, S.T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. J. Differential Geom. 78(3), 369–428 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Fu, J.X., Yau, S.T.: A Monge-Ampère type equation motivated by string theory. Comm. Anal. Geom. 15(1), 29–76 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Garcia-Fernandez, M.: Lectures on the Strominger system. Travaux mathématiques Vol. XXIV, 7-61, Luxembourg (2016)

  13. 13.

    Graña, M., Minasian, R., Petrini, M., Tomasiello, A.: Generalized structures of \(N=1\) vacua. JHEP 11, 020 (2005)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Hull, C.: Compactifications of the Heterotic Superstring. Phys. Lett. B 178(4), 357–364 (1986)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ladyzenskaja, O.A., Solonnikov, V. A., Uraltseva, N.N. Linear and Quasi-linear Equations of Parabolic Type, Nauka, Moscow, 1967 [Russian]; English transl., Translations of Mathematical Monographs Vol. 23, AMS, Providence, RI (1968)

  16. 16.

    Lotay, J.: Geometric flows of \(G_2\) structures. Lectures and Surveys on \(G_2\)-Manifolds and Related Topics, 113–140, Fields Inst. Commun., 84, Springer (2020)

  17. 17.

    Lotay, J., Wei, Y.: Laplacian flow for closed \(G_2\) structures: real analyticity. Comm. Anal. Geom. 27, 73–109 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Phong, D.H., Picard, S., Zhang, X.W.: Geometric flows and Strominger systems. Math. Z. 288(1–2), 101–113 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Phong, D.H., Picard, S., Zhang, X.W.: The anomaly flow and the Fu-Yau equation. Ann. PDE 4(2), Paper No. 13, 60 (2018)

  20. 20.

    Phong, D.H., Picard, S., Zhang, X.W.: A flow of conformally balanced metrics with Kähler fixed points. Math. Ann. 374(3–4), 2005–2040 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Phong, D.H., Picard, S., Zhang, X.W.: The Anomaly flow on unimodular Lie groups, Advances in Complex Geometry, 217–237, Contemporary Mathematics, vol. 735. AMS, Providence, RI (2019)

    Google Scholar 

  22. 22.

    Phong, D.H., Picard, S., Zhang, X.W.: The Fu-Yau equation with negative slope parameter. Invent. Math. 209(2), 541–576 (2017)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Phong, D.H., Picard, S., Zhang, X.W.: Anomaly flows. Comm. Anal. Geom. 26(4), 955–1008 (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Phong, D.H., Sesum, N., Sturm, J.: Multiplier ideal sheaves and the Kähler-Ricci flow. Comm. Anal. Geom. 15(3), 613–632 (2007)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Phong, D.H., Song, J., Sturm, J., Weinkove, B.: On the convergence of the modified Kähler-Ricci flow and solitons. Comment. Math. Helv. 86(1), 91–112 (2011)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pujia, M., Ugarte, L.: The Anomaly flow on nilmanifolds. arXiv:2004.06744

  27. 27.

    Streets, J., Tian, G.: Hermitian curvature flow. J. Eur. Math. Soc. 13(3), 601–634 (2011)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Strominger, A.: Superstrings with torsion. Nuclear Phys. B 274(2), 253–284 (1986)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Tomasiello, A.: Generalized structures of ten-dimensional supersymmetric solutions. JHEP 03, 073 (2012)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Tseng, L.S., Yau, S.T.: Cohomology and Hodge theory on symplectic manifolds: I. J. Differential Geom. 91(3), 383–416 (2012)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Tseng, L.S., Yau, S.T.: Cohomology and Hodge theory on symplectic manifolds: II. J. Differential Geom. 91(3), 417–443 (2012)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Tseng, L.S., Yau, S.T.: Generalized cohomologies and supersymmetry. Comm. Math. Phys. 326(3), 875–885 (2014)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Ustinovskiy, Y.: Hermitian curvature flow and curvature positivity conditions. Princeton University, PhD Thesis (2018)

  34. 34.

    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31(3), 339–411 (1978)

  35. 35.

    Zhang, X., Zhang, X.W.: Regularity estimates for solutions to complex Monge-Ampère equations on Hermitian manifolds. J. Funct. Anal. 260(7), 2004–2026 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Sebastien Picard.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Work supported in part by the National Science Foundation Grants DMS-1855947 and DMS-1809582.

Communicated by Ngaiming Mok.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fei, T., Phong, D.H., Picard, S. et al. Estimates for a geometric flow for the Type IIB string. Math. Ann. (2021).

Download citation