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.

show all references

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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