American Institute of Mathematical Sciences

September  2012, 17(6): 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

A Monge-Ampère type fully nonlinear equation on Hermitian manifolds

 1 Department of Mathematics, The Ohio State University, Columbus, OH 43210 2 Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, United States

Received  March 2012 Revised  March 2012 Published  May 2012

We study a fully nonlinear equation of complex Monge-Ampère type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.
Citation: Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991
References:
 [1] T. Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, (French) Bull. Sci. Math. (2), 102 (1978), 63-95. [2] X. Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Notices, 2000, 607-623. [3] X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852. [4] P. Cherrier, Équations de Monge-Ampère sur les variétés hermitiennes compactes, Bull. Sci. Math. (2), 111 (1987), 343-385. [5] S. K. Donaldson, Moment maps and diffeomorphisms. Sir Michael Atiyah: A great mathematician of the twentieth century, Asian J. Math., 3 (1999), 1-15. [6] H. Fang, M. Lai and X. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220. [7] B. Guan and Q. Li, Complex Monge-Ampère equations and totally real submanifolds, Adv. Math., 225 (2010), 1185-1223. doi: 10.1016/j.aim.2010.03.019. [8] A. Hanani, Équations du type de Monge-Ampère sur les variétés hermitiennes compactes, (French) [Monge-Ampère equations on compact Hermitian manifolds], J. Funct. Anal., 137 (1996), 49-75. doi: 10.1006/jfan.1996.0040. [9] A. Hanani, Une généralisation de l'équation de Monge-Ampère sur les variétés hermitiennes compactes, (French) [A generalization of the Monge-Ampère equation on compact Hermitian manifolds], Bull. Sci. Math., 120 (1996), 215-252. [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] V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40. [12] 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. [13] B. Weinkove, Convergence of the J-flow on Kähler surfaces, Comm. Anal. Geom., 12 (2004), 949-965. [14] B. Weinkove, On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom., 73 (2006), 351-358. [15] 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. [16] X. Zhang, A priori estimate for complex Monge-Ampère equation on Hermitian manifolds, Int. Math. Res. Notices, 2010, 3814-3836.

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References:
 [1] T. Aubin, Équations du type Monge-Ampère sur les variétés kählériennes compactes, (French) Bull. Sci. Math. (2), 102 (1978), 63-95. [2] X. Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Notices, 2000, 607-623. [3] X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852. [4] P. Cherrier, Équations de Monge-Ampère sur les variétés hermitiennes compactes, Bull. Sci. Math. (2), 111 (1987), 343-385. [5] S. K. Donaldson, Moment maps and diffeomorphisms. Sir Michael Atiyah: A great mathematician of the twentieth century, Asian J. Math., 3 (1999), 1-15. [6] H. Fang, M. Lai and X. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220. [7] B. Guan and Q. Li, Complex Monge-Ampère equations and totally real submanifolds, Adv. Math., 225 (2010), 1185-1223. doi: 10.1016/j.aim.2010.03.019. [8] A. Hanani, Équations du type de Monge-Ampère sur les variétés hermitiennes compactes, (French) [Monge-Ampère equations on compact Hermitian manifolds], J. Funct. Anal., 137 (1996), 49-75. doi: 10.1006/jfan.1996.0040. [9] A. Hanani, Une généralisation de l'équation de Monge-Ampère sur les variétés hermitiennes compactes, (French) [A generalization of the Monge-Ampère equation on compact Hermitian manifolds], Bull. Sci. Math., 120 (1996), 215-252. [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] V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40. [12] 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. [13] B. Weinkove, Convergence of the J-flow on Kähler surfaces, Comm. Anal. Geom., 12 (2004), 949-965. [14] B. Weinkove, On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy, J. Differential Geom., 73 (2006), 351-358. [15] 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. [16] X. Zhang, A priori estimate for complex Monge-Ampère equation on Hermitian manifolds, Int. Math. Res. Notices, 2010, 3814-3836.
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