Skip to main content
SpringerLink
Log in

The Boundary Case for the Supercritical Deformed Hermitian–Yang–Mills Equation

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, we shall study the weak solution to the supercritical deformed Hermitian–Yang–Mills equation in the boundary case.

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

Access this article

Log in via an institution

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

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Berman, R.: From Monge–Ampère equations to envelopes and geodesic rays in the zero temperature limit. Math. Z. 291, 365–394 (2019)

    Article  MathSciNet  Google Scholar 

  2. Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Amm. Sci. Ecole Norm. Sup. 37(4), 45–76 (2004)

    Article  MathSciNet  Google Scholar 

  3. Chen, G.: The J-equation and the supercritical deformed Hermitian–Yang–Mills equation. Invent. Math. 225, 529–602 (2021)

    Article  MathSciNet  Google Scholar 

  4. Chen, Y.-Z., Wu, L.-C.: Second oder elliptic equations and elliptic systems. Am. Math. Soc. (1998). https://doi.org/10.1090/mmono/174

    Article  Google Scholar 

  5. Collins, T.C., Jacob, A., Yau, S.-T.: \((1,1)\) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions. Camb. J. Math. 8(2), 407–452 (2020)

    Article  MathSciNet  Google Scholar 

  6. Demailly, J.-P., Paun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. 159, 1247–1274 (2004)

    Article  MathSciNet  Google Scholar 

  7. Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)

    Article  MathSciNet  Google Scholar 

  8. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009)

    Article  Google Scholar 

  9. Fang, H., Lai, M.-J., Ma, X.-N.: On a class of fully nonlinear flows in Kähler geometry. J. Reine Angew. Math. 653, 189–220 (2011)

    MathSciNet  Google Scholar 

  10. Fu, J.-X., Yau, S.-T., Zhang, D.-K.: A deformed Hermitian Yang-Mills Flow. To appear in J. Differ. Geom

  11. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer-Verlag, Berlin-New York (1983)

    Google Scholar 

  12. Guan, B.: Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J. 163, 1491–1524 (2014)

    Article  MathSciNet  Google Scholar 

  13. Guedj, V., Zeriahi, A.: Degenerate Complex Monge–Ampère Equations. EMS Tracts in Mathematics, vol. 26. GmbH, Leipzig (2017)

    Book  Google Scholar 

  14. Guo, B., Phong, D.: On \(L^\infty \) estimates for fully nonlinear partial differential equations on Hermitian manifolds. arXiv:2204.12549

  15. Guo, B., Phong, D.: Uniform entropy and energy bounds for fully non-linear equations. arXiv:2207.08983

  16. Guo, B., Phong, D., Tong, F.: On \(L^{\infty }\) estimates for complex Monge–Ampère equations. Ann. Math. 198(1), 393–418 (2023)

    Article  MathSciNet  Google Scholar 

  17. Guo, B., Phong, D., Tong, F., Wang, C.: On \(L^{\infty }\) estimates for Monge–Ampère and Hessian equations on nef classes. arXiv:2111.14186

  18. Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MathSciNet  Google Scholar 

  19. Hörmander, L.: An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton, NJ (1973)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Huisken, G., Sinestrari, C.: Convexity estimate for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)

    Article  MathSciNet  Google Scholar 

  22. Jacob, A., Yau, S.T.: A special Lagrangian type equation for holomorphic line bundles. Math. Ann. 369(1–2), 869–898 (2017)

    Article  MathSciNet  Google Scholar 

  23. Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations. Izvestiya Ross. Akad. Nauk. SSSR 46(3), 487–523 (1982)

    Google Scholar 

  24. Leung, N., Yau, S. T., Zaslow, E.: From special Lagrangian to Hermitian–Yang–Mills via Fourier-Mukai transform. AMS/IP Stud. Adv. Math. 23, 209–225. Amer. Math. Soc., Providence, RI (2001)

  25. Mariño, M., Minasian, R., Moore, G., Strominger, A.: Nonlinear instantons from supersymmetric \(p\)-branes. J. High Energy Phys. 1, 32 (2000)

    MathSciNet  Google Scholar 

  26. Strominger, A., Yau, S.T., Zaslow, E.: Mirror symmetry is \(T\)-duality. Nuclear Phys. B 479, 243–259 (1996)

    Article  MathSciNet  Google Scholar 

  27. Sui, Z., Sun, W.: On \(L^\infty \) estimate for complex Hessian quotient equations on compact Kähler manifolds. J. Geom. Anal. 33, 165 (2023)

    Article  Google Scholar 

  28. Sun, W.: The boundary case for complex Monge–Ampère type equations. arXiv:2312.06120

  29. Sun, W.: The weak solutions to complex Hessian equations. Calc. Var. PDE 63, 57 (2024)

    Article  MathSciNet  Google Scholar 

  30. Sun, W.: The boundary case of the \(J\)-flow. arXiv:2306.11794

  31. Sun, W.: Viscosity solutions to uniformly elliptic complex equations. arXiv:2306.00651

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

    Article  MathSciNet  Google Scholar 

  33. Tian, G.: On Kähler–Einstein metrics on certain Kähler manifolds with \(C_1 (M) > 0\). Invent. Math. 89(2), 225–246 (1987)

    Article  MathSciNet  Google Scholar 

  34. Tosatti, V., Wang, Y., Weinkove, B., Yang, X.: \(C^{2,\alpha }\) estimates for nonlinear elliptic equations in complex and almost complex geometry. Calc. Var. PDE 54(1), 431–453 (2015)

    Article  Google Scholar 

  35. Wang, D., Yuan, Y.: Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions. Am. J. Math. 136, 481–499 (2014)

    Article  MathSciNet  Google Scholar 

  36. Wang, J., Wang, X.-J., Zhou, B.: A priori estimates for the complex Monge–Ampère equations. Peking Math. J. 4, 143–157 (2021)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

The author wish to thank Gao Chen, Bo Guan and Rirong Yuan for helpful discussions. The author is supported by National Natural Science Foundation of China (12371207) and a start-up grant from ShanghaiTech University (2018F0303-000-04).

Author information

Authors and Affiliations

  1. Institute of Mathematical Sciences, ShanghaiTech University, Shanghai, China

    Wei Sun

Authors
  1. Wei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Sun.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, W. The Boundary Case for the Supercritical Deformed Hermitian–Yang–Mills Equation. J Geom Anal 34, 177 (2024). https://doi.org/10.1007/s12220-024-01636-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-024-01636-3

Keywords

Mathematics Subject Classification

Search

Navigation