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September  2017, 16(5): 1553-1570. doi: 10.3934/cpaa.2017074

On uniform estimate of complex elliptic equations on closed Hermitian manifolds

Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  July 2015 Revised  January 2017 Published  May 2017

Fund Project: The author is supported by China Postdoctoral Science Foundation (Grant No. 2015M571478) and National Natural Science Foundation of China (Grant No. 11501119).

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

Citation: Wei Sun. On uniform estimate of complex elliptic equations on closed Hermitian manifolds. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1553-1570. doi: 10.3934/cpaa.2017074
References:
[1]

Z. Blocki, On uniform estimate in Calabi-Yau theorem, Sci. China Ser. A, 48 (2005), 244-247.  doi: 10.1007/BF02884710.

[2]

X. X. Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Notices, 2000 (2000), 607-623.  doi: 10.1155/S1073792800000337.

[3]

P. Cherrier, Equations de Monge-Ampére sur les variétés hermitiennes compactes, Bull. Sci. Math., 111 (1987), 343-385. 

[4]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Am. J. Math. to appear.

[5]

S. K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 3 (1999), 1-16.  doi: 10.4310/AJM.1999.v3.n1.a1.

[6]

H. FangM. J. Lai and X. N. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.  doi: 10.1515/CRELLE.2011.027.

[7]

L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech., 8 (1959), 957-965. 

[8]

B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc.Var. PDE, 54 (2015), 901-916.  doi: 10.1007/s00526-014-0810-1.

[9]

Z. L. HouX. 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.

[10]

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.

[11]

W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, J. Geom. Anal., 26 (2016), 2459-2473.  doi: 10.1007/s12220-015-9634-2.

[12]

W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds Ⅱ: $L^∞ $ estimate, Comm. Pure Appl. Math., 70 (2017), 172-199.  doi: 10.1002/cpa.21652.

[13]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. to appear.

[14]

V. TosattiY. WangB. Weinkove and X.-K. Yang, $C^{2,α} $ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. PDE, 54 (2015), 431-453.  doi: 10.1007/s00526-014-0791-0.

[15]

V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampére equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.  doi: 10.4310/AJM.2010.v14.n1.a3.

[16]

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

[17]

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

[18]

D. K. Zhang, Hessian equations on closed Hermitian manifolds, preprint, arXiv: 1501.03553.

show all references

References:
[1]

Z. Blocki, On uniform estimate in Calabi-Yau theorem, Sci. China Ser. A, 48 (2005), 244-247.  doi: 10.1007/BF02884710.

[2]

X. X. Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Notices, 2000 (2000), 607-623.  doi: 10.1155/S1073792800000337.

[3]

P. Cherrier, Equations de Monge-Ampére sur les variétés hermitiennes compactes, Bull. Sci. Math., 111 (1987), 343-385. 

[4]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Am. J. Math. to appear.

[5]

S. K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 3 (1999), 1-16.  doi: 10.4310/AJM.1999.v3.n1.a1.

[6]

H. FangM. J. Lai and X. N. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.  doi: 10.1515/CRELLE.2011.027.

[7]

L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech., 8 (1959), 957-965. 

[8]

B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc.Var. PDE, 54 (2015), 901-916.  doi: 10.1007/s00526-014-0810-1.

[9]

Z. L. HouX. 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.

[10]

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.

[11]

W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, J. Geom. Anal., 26 (2016), 2459-2473.  doi: 10.1007/s12220-015-9634-2.

[12]

W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds Ⅱ: $L^∞ $ estimate, Comm. Pure Appl. Math., 70 (2017), 172-199.  doi: 10.1002/cpa.21652.

[13]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. to appear.

[14]

V. TosattiY. WangB. Weinkove and X.-K. Yang, $C^{2,α} $ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. PDE, 54 (2015), 431-453.  doi: 10.1007/s00526-014-0791-0.

[15]

V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampére equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.  doi: 10.4310/AJM.2010.v14.n1.a3.

[16]

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

[17]

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

[18]

D. K. Zhang, Hessian equations on closed Hermitian manifolds, preprint, arXiv: 1501.03553.

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