SpringerLink
Log in

The Second Hessian Type Equation on Almost Hermitian Manifolds

  • Research Article
  • Published:
Frontiers of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we derive the second order estimate to the 2nd Hessian type equation on a compact almost Hermitian manifold.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chu J., Tosatti V., Weinkove B., The Monge–Ampère equation for non-integrable almost complex structures. J. Eur. Math. Soc. (JEMS), 2019, 21(7): 1949–1984

    Article  MathSciNet  Google Scholar 

  2. Chu J., Huang L., Zhang J., Fully non-linear elliptic equations on compact almost Hermitian manifolds. Calc. Var. Partial Differential Equations, 2023, 62(3): Paper No. 105, 34 pp.

  3. Chu J., Huang L., Zhu X., The Fu–Yau Equation in higher dimensions. Peking Math. J., 2019, 2(1): 71–97

    Article  MathSciNet  Google Scholar 

  4. Chu J., Huang L., Zhu X., The Fu–Yau equation on compact astheno-Kähler manifolds. Adv. Math., 2019, 346: 908–945

    Article  MathSciNet  Google Scholar 

  5. De Bartolomeis P., Tomassini A., On solvable generalized Calabi–Yau manifolds. Ann. Inst. Fourier (Grenoble), 2006, 56(5): 1281–1296

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Fu J., Yau S.-T., A Monge–Ampère-type equation motivated by string theory. Comm. Anal. Geom., 2007, 15(1): 29–75

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Greene B.R., Shapere A., Vafa C., Yau S.-T., Stringy cosmic strings and noncompact Calabi–Yau manifolds. Nuclear Phys. B, 1990, 337(1): 1–36

    Article  MathSciNet  Google Scholar 

  10. Harvey F.R., Lawson H.B., Potential theory on almost complex manifolds. Ann. Inst. Fourier (Grenoble), 2015, 65(1): 171–210

    Article  MathSciNet  Google Scholar 

  11. Hitchin N., Generalized Calabi–Yau manifolds. Q. J. Math., 2003, 54(3): 281–308

    Article  MathSciNet  Google Scholar 

  12. Horn R.A., Johnson C.R., Matrix Analysis, Second Edition. Cambridge: Cambridge University Press, 2013

    Google Scholar 

  13. Hou Z., Complex Hessian equation on Kähler manifold. Int. Math. Res. Not. IMRN, 2009, 16: 3098–3111

    Article  Google Scholar 

  14. Hou Z., Ma X., Wu D., A second order estimate for complex Hessian equations on a compact Kähler manifold. Math. Res. Lett., 2010, 17(3): 547–561

    Article  MathSciNet  Google Scholar 

  15. Huang L., Zhang J., Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds. Math. Z., 2023, 303(2): Paper No. 36, 24 pp.

  16. Lin M., Trudinger N.S., On some inequalities for elementary symmetric functions. Bull. Aust. Math. Soc., 1994, 50: 317–326

    Article  MathSciNet  Google Scholar 

  17. Phong D.H., Picard S., Zhang X., A second order estimate for general complex Hessian equations. Anal. PDE, 2016, 9(7): 1693–1709

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Phong D.H., Picard S., Zhang, X., On estimates for the Fu–Yau generalization of a Strominger system. J. Reine Angew. Math., 2019, 751: 243–274

    Article  MathSciNet  Google Scholar 

  20. Phong D.H., Picard S., Zhang X., Fu–Yau Hessian equations. J. Differential Geom., 2021, 118(1): 147–187

    Article  MathSciNet  Google Scholar 

  21. Streets J., Tian G., Generalized Kähler geometry and the pluriclosed flow. Nuclear Phys. B, 2012, 858(2): 366–376

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Székelyhidi G., Fully non-linear elliptic equations on compact Hermitian manifolds. J. Differential Geom., 2018, 109(2): 337–378

    Article  MathSciNet  Google Scholar 

  24. Székelyhidi G., Tosatti V., Weinkove B., Gauduchon metrics with prescribed volume form. Acta Math., 2017, 219(1): 181–211

    Article  MathSciNet  Google Scholar 

  25. Zhang D., Hessian equations on closed Hermitian manifolds. Pacific J. Math., 2017, 291(2): 485–510

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author would like to thank his advisor G. Tian for encouragement and support. He also thanks V. Tosatti and B. Weinkove for their collaboration. The first and third authors are partially supported by National Key R&D Program of China (No. 2023YFA1009900). The first author is partially supported by NSFC (No. 12271008) and the Fundamental Research Funds for the Central Universities, Peking University. The third author is partially supported by National Key R&D Program of China (No. 2020YFA0712800) and NSFC (No. 12271009).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Bei**g, 100871, China

    Jianchun Chu & **aohua Zhu

  2. School of Mathematical Sciences, **amen University, **amen, 361005, China

    Liding Huang

  3. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    Liding Huang

  4. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Bei**g, 100190, China

    Jianchun Chu

Authors
  1. Jianchun Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liding Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. **aohua Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liding Huang.

Ethics declarations

Conflict of Interest X.H. Zhu is a member of Editorial Board of Frontiers of Mathematics and was not involved in the editorial review or the decision to publish this article. The authors declare no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chu, J., Huang, L. & Zhu, X. The Second Hessian Type Equation on Almost Hermitian Manifolds. Front. Math (2024). https://doi.org/10.1007/s11464-024-0018-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11464-024-0018-4

Keywords

MSC2020

Search

Navigation