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

A Monge-Ampère type fully nonlinear equation on Hermitian manifolds

Bo Guan 1, and  Qun Li 2,

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.
Keywords: a priori estimates., fully nonlinear equations, Hermitian manifolds, Complex Monge-Ampère equations.
Mathematics Subject Classification: 58J05, 58J32, 32W20, 35J25, 53C5.
Citation: Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & 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.  Google Scholar

[2]

X. Chen, On the lower bound of the Mabuchi energy and its application,, Int. Math. Res. Notices, 2000 (): 607.   Google Scholar

[3]

X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852.  Google Scholar

[4]

P. Cherrier, Équations de Monge-Ampère sur les variétés hermitiennes compactes, Bull. Sci. Math. (2), 111 (1987), 343-385.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.  Google Scholar

[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.  Google Scholar

[13]

B. Weinkove, Convergence of the J-flow on Kähler surfaces, Comm. Anal. Geom., 12 (2004), 949-965.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

X. Zhang, A priori estimate for complex Monge-Ampère equation on Hermitian manifolds,, Int. Math. Res. Notices, 2010 (): 3814.   Google Scholar

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.  Google Scholar

[2]

X. Chen, On the lower bound of the Mabuchi energy and its application,, Int. Math. Res. Notices, 2000 (): 607.   Google Scholar

[3]

X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852.  Google Scholar

[4]

P. Cherrier, Équations de Monge-Ampère sur les variétés hermitiennes compactes, Bull. Sci. Math. (2), 111 (1987), 343-385.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

V. Tosatti and B. Weinkove, Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.  Google Scholar

[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.  Google Scholar

[13]

B. Weinkove, Convergence of the J-flow on Kähler surfaces, Comm. Anal. Geom., 12 (2004), 949-965.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

X. Zhang, A priori estimate for complex Monge-Ampère equation on Hermitian manifolds,, Int. Math. Res. Notices, 2010 (): 3814.   Google Scholar

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