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A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation
Second order estimates for complex Hessian equations on Hermitian manifolds
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
R. Kobayashi,
Kähler-Einstein metric on an open algebraic manifold, Osaka J. Math., 21 (1984), 399-418.
|
[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. |
[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. |
[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. |
[16] |
D. H. Phong, S. Picard and X. W. Zhang, Fu-Yau Hessian equations, preprint, arXiv: 1801.09842. |
[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. |
[18] |
G. Székelyhidi,
Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom., 109 (2018), 337-378.
doi: 10.4310/jdg/1527040875. |
[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. |
[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. |
[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. |
[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. |
[23] |
V. Tosatti and B. Weinkove, The complex Monge-Ampère equation with a gradient term, preprint, arXiv: 1906.10034. |
[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. |
[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. |
[26] |
D. K. Zhang,
Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.
doi: 10.2140/pjm.2017.291.485. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
R. Kobayashi,
Kähler-Einstein metric on an open algebraic manifold, Osaka J. Math., 21 (1984), 399-418.
|
[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. |
[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. |
[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. |
[16] |
D. H. Phong, S. Picard and X. W. Zhang, Fu-Yau Hessian equations, preprint, arXiv: 1801.09842. |
[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. |
[18] |
G. Székelyhidi,
Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom., 109 (2018), 337-378.
doi: 10.4310/jdg/1527040875. |
[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. |
[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. |
[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. |
[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. |
[23] |
V. Tosatti and B. Weinkove, The complex Monge-Ampère equation with a gradient term, preprint, arXiv: 1906.10034. |
[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. |
[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. |
[26] |
D. K. Zhang,
Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.
doi: 10.2140/pjm.2017.291.485. |
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