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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 & 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. Google Scholar [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. Google Scholar [3] P. Cherrier, Equations de Monge-Ampére sur les variétés hermitiennes compactes, Bull. Sci. Math., 111 (1987), 343-385. Google Scholar [4] S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Am. J. Math. to appear.Google Scholar [5] S. K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 3 (1999), 1-16. doi: 10.4310/AJM.1999.v3.n1.a1. Google Scholar [6] H. Fang, M. 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. Google Scholar [7] L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech., 8 (1959), 957-965. Google Scholar [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. Google Scholar [9] 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 [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. to appear.Google Scholar [14] V. Tosatti, Y. Wang, B. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] D. K. Zhang, Hessian equations on closed Hermitian manifolds, preprint, arXiv: 1501.03553.Google Scholar

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##### References:
 [1] Z. Blocki, On uniform estimate in Calabi-Yau theorem, Sci. China Ser. A, 48 (2005), 244-247. doi: 10.1007/BF02884710. Google Scholar [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. Google Scholar [3] P. Cherrier, Equations de Monge-Ampére sur les variétés hermitiennes compactes, Bull. Sci. Math., 111 (1987), 343-385. Google Scholar [4] S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Am. J. Math. to appear.Google Scholar [5] S. K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 3 (1999), 1-16. doi: 10.4310/AJM.1999.v3.n1.a1. Google Scholar [6] H. Fang, M. 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. Google Scholar [7] L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech., 8 (1959), 957-965. Google Scholar [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. Google Scholar [9] 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 [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differential Geom. to appear.Google Scholar [14] V. Tosatti, Y. Wang, B. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] D. K. Zhang, Hessian equations on closed Hermitian manifolds, preprint, arXiv: 1501.03553.Google Scholar
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