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June  2021, 41(6): 2619-2633. doi: 10.3934/dcds.2020377

Second order estimates for complex Hessian equations on Hermitian manifolds

Weisong Dong 1, and  Chang Li 2,,

1. 

School of Mathematics, Tianjin University, Tianjin 300354, China
2. 

Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Chang Li

Received  April 2020 Revised  September 2020 Published  November 2020

We derive second order estimates for $ \chi $-plurisubharmonic solutions of complex Hessian equations with right hand side depending on the gradient on compact Hermitian manifolds.

Keywords: Complex Hessian equations, Second order estimates, Hermitian manifolds.
Mathematics Subject Classification: Primary: 35J15, 53C55; Secondary: 58J05, 35B45.
Citation: Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377
References:
[1]

S. Y. Cheng and S. T. Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation, Comm. Pure Appl. Math., 33 (1980), 507-544.  doi: 10.1002/cpa.3160330404.  Google Scholar

[2]

J. C. Chu, L. D. Huang and X. H. Zhu, The Fu-Yau equation in higher dimensions, Peking Math. J., 2 (2019), 71-97.  doi: 10.1007/s42543-019-00016-z.  Google Scholar

[3]

J. C. Chu, L. D. Huang and X. H. Zhu, The Fu-Yau equation on compact astheno-Kähler manifolds, Adv. Math., 346 (2019), 908-945.  doi: 10.1016/j.aim.2019.02.006.  Google Scholar

[4]

J. C. Chu, L. D. Huang and X. H. Zhu, The 2-nd Hessian type equation on almost Hermitian manifolds, preprint, arXiv: 1707.04072. Google Scholar

[5]

J. C. Chu, V. Tosatti and B. Weinkove, The Monge-Ampère equation for non-integrable almost complex structures, J. Eur. Math. Soc., 21 (2019), 1949-1984.  doi: 10.4171/JEMS/878.  Google Scholar

[6]

S. Dinew and S. Kołodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math., 139 (2017), 403-415.  doi: 10.1353/ajm.2017.0009.  Google Scholar

[7]

A. Fino, Y. Y. Li, S. Salamon and L. Vezzoni, The Calabi-Yau equation on 4-manifolds over 2-tori, Trans. Amer. Math. Soc., 365 (2013), 1551-1575.  doi: 10.1090/S0002-9947-2012-05692-3.  Google Scholar

[8]

J. X. Fu and S. T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation, J. Differential Geom., 78 (2008), 369-428.  doi: 10.4310/jdg/1207834550.  Google Scholar

[9]

B. Guan and H. M. Jiao, Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 54 (2015), 2693-2712.  doi: 10.1007/s00526-015-0880-8.  Google Scholar

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P. F. Guan, C. Y. Ren and Z. Z. Wang, Global $C^2$-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68 (2015), 1287-1325.  doi: 10.1002/cpa.21528.  Google Scholar

[11]

Z. L. Hou, X. N. Ma and D. M. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.  doi: 10.4310/MRL.2010.v17.n3.a12.  Google Scholar

[12]

R. Kobayashi, Kähler-Einstein metric on an open algebraic manifold, Osaka J. Math., 21 (1984), 399-418.   Google Scholar

[13]

C. Li and L. M. Shen, The complex Hessian equations with gradient terms on Hermitian manifolds, J. Differential Equations, 269 (2020), 6293-6310.  doi: 10.1016/j.jde.2020.04.037.  Google Scholar

[14]

Y. Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampere type, Comm. Pure Appl. Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[15]

D. H. Phong, S. Picard and X. W. Zhang, A second order estimate for general complex Hessian equations, Anal. PDE, 9 (2016), 1693-1709.  doi: 10.2140/apde.2016.9.1693.  Google Scholar

[16]

D. H. Phong, S. Picard and X. W. Zhang, Fu-Yau Hessian equations, preprint, arXiv: 1801.09842. Google Scholar

[17]

J. Song and B. Weinkove, On the convergence and singularities of the J-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math., 61 (2008), 210-229.  doi: 10.1002/cpa.20182.  Google Scholar

[18]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom., 109 (2018), 337-378.  doi: 10.4310/jdg/1527040875.  Google Scholar

[19]

G. Tian, On the existence of solutions of a class of Monge-Ampère equations, Acta Math. Sinica (N.S.), 4 (1988), 250-265.  doi: 10.1007/BF02560581.  Google Scholar

[20]

G. Tian and S. T. Yau, Complete Kähler manifolds with zero Ricci curvature. Ⅰ, J. Amer. Math. Soc., 3 (1990), 579-609.  doi: 10.2307/1990928.  Google Scholar

[21]

G. Tian and S. T. Yau, Complete Kähler manifolds with zero Ricci curvature. Ⅱ, Invent. Math., 106 (1991), 27-60.  doi: 10.1007/BF01243902.  Google Scholar

[22]

V. Tosatti and B. Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc., 23 (2010), 1187-1195.  doi: 10.1090/S0894-0347-2010-00673-X.  Google Scholar

[23]

V. Tosatti and B. Weinkove, The complex Monge-Ampère equation with a gradient term, preprint, arXiv: 1906.10034. Google Scholar

[24]

S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. Ⅰ, Comm. Pure Appl. Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.  Google Scholar

[25]

R. R. Yuan, On a class of fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds, Canad. J. Math., 70 (2018), 943-960.  doi: 10.4153/CJM-2017-015-9.  Google Scholar

[26]

D. K. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.  doi: 10.2140/pjm.2017.291.485.  Google Scholar

show all references

References:
[1]

S. Y. Cheng and S. T. Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation, Comm. Pure Appl. Math., 33 (1980), 507-544.  doi: 10.1002/cpa.3160330404.  Google Scholar

[2]

J. C. Chu, L. D. Huang and X. H. Zhu, The Fu-Yau equation in higher dimensions, Peking Math. J., 2 (2019), 71-97.  doi: 10.1007/s42543-019-00016-z.  Google Scholar

[3]

J. C. Chu, L. D. Huang and X. H. Zhu, The Fu-Yau equation on compact astheno-Kähler manifolds, Adv. Math., 346 (2019), 908-945.  doi: 10.1016/j.aim.2019.02.006.  Google Scholar

[4]

J. C. Chu, L. D. Huang and X. H. Zhu, The 2-nd Hessian type equation on almost Hermitian manifolds, preprint, arXiv: 1707.04072. Google Scholar

[5]

J. C. Chu, V. Tosatti and B. Weinkove, The Monge-Ampère equation for non-integrable almost complex structures, J. Eur. Math. Soc., 21 (2019), 1949-1984.  doi: 10.4171/JEMS/878.  Google Scholar

[6]

S. Dinew and S. Kołodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math., 139 (2017), 403-415.  doi: 10.1353/ajm.2017.0009.  Google Scholar

[7]

A. Fino, Y. Y. Li, S. Salamon and L. Vezzoni, The Calabi-Yau equation on 4-manifolds over 2-tori, Trans. Amer. Math. Soc., 365 (2013), 1551-1575.  doi: 10.1090/S0002-9947-2012-05692-3.  Google Scholar

[8]

J. X. Fu and S. T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation, J. Differential Geom., 78 (2008), 369-428.  doi: 10.4310/jdg/1207834550.  Google Scholar

[9]

B. Guan and H. M. Jiao, Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 54 (2015), 2693-2712.  doi: 10.1007/s00526-015-0880-8.  Google Scholar

[10]

P. F. Guan, C. Y. Ren and Z. Z. Wang, Global $C^2$-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68 (2015), 1287-1325.  doi: 10.1002/cpa.21528.  Google Scholar

[11]

Z. L. Hou, X. N. Ma and D. M. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.  doi: 10.4310/MRL.2010.v17.n3.a12.  Google Scholar

[12]

R. Kobayashi, Kähler-Einstein metric on an open algebraic manifold, Osaka J. Math., 21 (1984), 399-418.   Google Scholar

[13]

C. Li and L. M. Shen, The complex Hessian equations with gradient terms on Hermitian manifolds, J. Differential Equations, 269 (2020), 6293-6310.  doi: 10.1016/j.jde.2020.04.037.  Google Scholar

[14]

Y. Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampere type, Comm. Pure Appl. Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[15]

D. H. Phong, S. Picard and X. W. Zhang, A second order estimate for general complex Hessian equations, Anal. PDE, 9 (2016), 1693-1709.  doi: 10.2140/apde.2016.9.1693.  Google Scholar

[16]

D. H. Phong, S. Picard and X. W. Zhang, Fu-Yau Hessian equations, preprint, arXiv: 1801.09842. Google Scholar

[17]

J. Song and B. Weinkove, On the convergence and singularities of the J-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math., 61 (2008), 210-229.  doi: 10.1002/cpa.20182.  Google Scholar

[18]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom., 109 (2018), 337-378.  doi: 10.4310/jdg/1527040875.  Google Scholar

[19]

G. Tian, On the existence of solutions of a class of Monge-Ampère equations, Acta Math. Sinica (N.S.), 4 (1988), 250-265.  doi: 10.1007/BF02560581.  Google Scholar

[20]

G. Tian and S. T. Yau, Complete Kähler manifolds with zero Ricci curvature. Ⅰ, J. Amer. Math. Soc., 3 (1990), 579-609.  doi: 10.2307/1990928.  Google Scholar

[21]

G. Tian and S. T. Yau, Complete Kähler manifolds with zero Ricci curvature. Ⅱ, Invent. Math., 106 (1991), 27-60.  doi: 10.1007/BF01243902.  Google Scholar

[22]

V. Tosatti and B. Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc., 23 (2010), 1187-1195.  doi: 10.1090/S0894-0347-2010-00673-X.  Google Scholar

[23]

V. Tosatti and B. Weinkove, The complex Monge-Ampère equation with a gradient term, preprint, arXiv: 1906.10034. Google Scholar

[24]

S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. Ⅰ, Comm. Pure Appl. Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.  Google Scholar

[25]

R. R. Yuan, On a class of fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds, Canad. J. Math., 70 (2018), 943-960.  doi: 10.4153/CJM-2017-015-9.  Google Scholar

[26]

D. K. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.  doi: 10.2140/pjm.2017.291.485.  Google Scholar

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