Skip to main content
SpringerLink
Log in

Hessian Equations of Krylov Type on Kähler Manifolds

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

Abstract

In this paper, we consider a Hessian equation with its structure as a combination of elementary symmetric functions on closed Kähler manifolds. We provide a sufficient and necessary condition for the solvability of this equation, which generalizes the results of Hessian equation and Hessian quotient equation. As a consequence, we can solve a complex Monge–Ampère type equation proposed by Chen in the case that one of the coefficients is negative. In this case, the equation is related the deformed Hermitian Yang-Mills equation. The key to our argument is a novel use of the special properties of the Hessian quotient operator \(\frac{\sigma _k}{\sigma _{k-1}}\).

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

Not applicable.

References

  1. Andrews, B.: Contraction of convex hypersurfaces in Euclidean space. Calc. Var. Partial Differ. Equ. 2, 151–171 (1994)

  2. Błocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier (Grenoble) 55(5), 1735–1756 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Błocki, Z.: On uniform estimate in Calabi–Yau theorem. Sci. China Ser. A 48, 244–247 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Błocki, Z.: On the uniform estimate in the Calabi–Yau theorem. II. Sci. China Math. 54, 1375–1377 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Buchdahl, N.: On compact Kähler surfaces. Ann. Inst. Fourier (Grenoble) 49(1), 287–302 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Nirenberg, L., Spruck, J.: Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)

    MathSciNet  MATH  Google Scholar 

  7. Chen, C.Q., Chen, L., Mei, X.Q., Xiang, N.: The Classical Neumann Problem for a class of mixed Hessian equations. Stud. Appl. Math. 148, 5–26 (2022)

    MathSciNet  Google Scholar 

  8. Chen, C.Q., Chen, L., Mei, X.Q., Xiang, N.: The Neumann problem for a class of mixed complex Hessian equations. Discret. Contin. Dyn. Syst. 42(9), 4203–4218 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Chen, L., Guo, X., He, Y.: A class of fully nonlinear equations arising in conformal geometry. Int. Math. Res. Not. 5, 3651–3676 (2022)

    MathSciNet  MATH  Google Scholar 

  10. Chen, L., Shang, A.G., Tu, Q.: A class of prescribed Weingarten curvature equations in Euclidean space. Commun. Partial Differ. Equ. 46(7), 1326–1343 (2021)

    MathSciNet  MATH  Google Scholar 

  11. Chen, X.X.: On the lower bound of the Mabuchi energy and its application. Int. Math. Res. Not. 12, 607–623 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Chen, X.X.: A new parabolic flow in Kähler manifolds. Commun. Anal. Geom. 12(4), 837–852 (2004)

    MATH  Google Scholar 

  13. Chou, K.S., Wang, X.J.: A variational theory of the Hessian equation. Commun. Pure Appl. Math. 54(9), 1029–1064 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Cherrier, P.: équations de Monge–Ampère sur les variétés Hermitiennes compactes. Bull. Sci. Math. (2) 111, 343–385 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Collins, T., Székelyhidi, G.: Convergence of the \(J\)-flow on toric manifolds. J. Differ. Geom. 107(1), 47–81 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Donaldson, S.: Moment maps and diffeomorphisms. Asian J. Math. 3(1), 1–15 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. 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  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Gauduchon, P.: Le théorème de l’excentricité nulle. C. R. Acad. Sci. Paris 285, 387–390 (1977)

    MathSciNet  MATH  Google Scholar 

  23. Guan, B., Li, Q.: Complex Monge–Ampère equations and totally real submanifolds. Adv. Math. 225(3), 1185–1223 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Guan, B., Li, Q.: A Monge–Ampère type fully nonlinear equation on Hermitian manifolds. Discret. Contin. Dyn. Syst. Ser. B 17, 1991–1999 (2012)

    MATH  Google Scholar 

  25. Guan, B., Sun, W.: On a class of fully nonlinear elliptic equations on Hermitian manifolds. Calc. Var. Partial Differ. Equ. 54(1), 901–916 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Guan, P.F., Zhang, X.W.: A class of curvature type equations. Pure Appl. Math Q. 17(3), 865–907 (2021). arXiv:1909.03645

    MathSciNet  MATH  Google Scholar 

  27. Hanani, A.: Équations du type de Monge-Ampère sur les variét’es Hermitiennes compactes. J. Funct. Anal. 137, 49–75 (1996)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 47(1), 75–108 (1983)

    MathSciNet  Google Scholar 

  32. Krylov, N.V.: On the general notion of fully nonlinear second order elliptic equation. Trans. Am. Math. Soc. 347(3), 857–895 (1995)

    MathSciNet  MATH  Google Scholar 

  33. Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  34. Pingali, Vamsi P.: A fully nonlinear generalized Monge–Ampère PDE on a torus. Electron. J. Differ. Equ. 211, 8 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Vamsi, P.: Pingali: a generalised Monge–Ampère equation. J. Partial Differ. Equ. 27(4), 333–346 (2014)

    MathSciNet  MATH  Google Scholar 

  36. Vamsi, P.: Pingali: a priori estimates for a generalized Monge–Ampère PDE on some compact Kähler manifolds. Complex Var. Elliptic Equ. 61(8), 1037–1051 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Pingali, Vamsi P.: The deformed Hermitian Yang-Mills equation on three-folds. arXiv:1910.01870v4 (2022)

  38. Phong, D.H., Picard, S., Zhang, X.W.: On estimates for the Fu–Yau generalization of a Strominger system. J. Reine Angew. Math. (Crelle’s Journal) 2019(751), 243–274 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. Phong, D.H., Tô, D.T.: Fully non-linear parabolic equations on compact Hermitian manifolds. Ann. Sci. Ec. Norm. Sup. 54(3), 793–829 (2021)

    MathSciNet  MATH  Google Scholar 

  41. Phong, D.H., Picard, S., Zhang, X.W.: Fu-Yau Hessian Equations. arXiv:1801.09842, to appear in J. Differential Geometry

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

    MathSciNet  MATH  Google Scholar 

  43. Song, J., Weinkove, B.: On the convergence and singularities of the \(J\)-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61, 210–229 (2008)

    MathSciNet  MATH  Google Scholar 

  44. Spruck, J.: Geometric aspects of the theory of fully nonlinear elliptic equations. Clay Math. Proc. 2, 283–309 (2005)

    MathSciNet  MATH  Google Scholar 

  45. Sun, W.: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds. J. Geom. Anal. 26(3), 2459–2473 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Sun, W.: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: \(L^{\infty }\) estimate. Commun. Pure Appl. Math. 70(1), 172–199 (2017)

    MathSciNet  MATH  Google Scholar 

  47. Sun, W.: Generalized complex Monge–Ampère type equations on closed Hermitian manifolds, preprint. arXiv:1412.8192

  48. Sun, W.: Parabolic Flow for Generalized complex Monge-Ampère type equations preprint, arXiv:1501.04255

  49. Tosatti, V., Weinkove, B.: Estimates for the complex Monge–Ampère equation on Hermitian and balanced manifolds. Asian J. Math. 14, 19–40 (2010)

    MathSciNet  MATH  Google Scholar 

  50. Tosatti, V., Weinkove, B.: The complex Monge–Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23, 1187–1195 (2010)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  52. Tsai, Y.L.: A type of parabolic flow with mixed Hessian on compact Kähler manifolds, preprint

  53. Weinkove, B.: Convergence of the \(J\)-flow on Kähler surfaces. Commun. Anal. Geom. 12(4), 949–965 (2004)

    MATH  Google Scholar 

  54. Weinkove, B.: The \(J\)-flow, the Mabuchi energy, the Yang-Mills flow and multiplier ideal sheaves. Doctoral dissertation, Columbia University, New York (2004)

    Google Scholar 

  55. Weinkove, B.: On the \(J\)-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differ. Geom. 73(2), 351–358 (2006)

    MathSciNet  MATH  Google Scholar 

  56. 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)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  58. Zhang, X.W.: A priori estimates for complex Monge–Ampère equation on Hermitian manifolds. Int. Math. Res. Not. 19, 3814–3836 (2010)

    MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express his gratitude to Dr. Tsai for sending us their manuscript [52].

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, 430062, People’s Republic of China

    Li Chen

Authors
  1. Li Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Chen.

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

Chen, L. Hessian Equations of Krylov Type on Kähler Manifolds. J Geom Anal 33, 333 (2023). https://doi.org/10.1007/s12220-023-01394-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-023-01394-8

Keywords

Mathematics Subject Classification

Search

Navigation