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 & 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.

[2]

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

[3]

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

[4]

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

[5]

S. K. Donaldson, Moment maps and diffeomorphisms. Sir Michael Atiyah: A great mathematician of the twentieth century,, Asian J. Math., 3 (1999), 1.

[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.

[7]

B. Guan and Q. Li, Complex Monge-Ampère equations and totally real submanifolds,, Adv. Math., 225 (2010), 1185. 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. 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.

[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. 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.

[12]

V. Tosatti and B. Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds,, J. Amer. Math. Soc., 23 (2010), 1187. 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.

[14]

B. Weinkove, On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy,, J. Differential Geom., 73 (2006), 351.

[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. 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.

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.

[2]

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

[3]

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

[4]

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

[5]

S. K. Donaldson, Moment maps and diffeomorphisms. Sir Michael Atiyah: A great mathematician of the twentieth century,, Asian J. Math., 3 (1999), 1.

[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.

[7]

B. Guan and Q. Li, Complex Monge-Ampère equations and totally real submanifolds,, Adv. Math., 225 (2010), 1185. 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. 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.

[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. 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.

[12]

V. Tosatti and B. Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds,, J. Amer. Math. Soc., 23 (2010), 1187. 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.

[14]

B. Weinkove, On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy,, J. Differential Geom., 73 (2006), 351.

[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. 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.

[1]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121

[2]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825

[3]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061

[4]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[5]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

[6]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705

[7]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069

[8]

Diego Maldonado. On interior \begin{document} $C^2$ \end{document}-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[9]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559

[10]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[11]

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

[12]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013

[13]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[14]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447

[15]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697

[16]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499

[17]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539

[18]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247

[19]

Feliz Minhós. Periodic solutions for some fully nonlinear fourth order differential equations. Conference Publications, 2011, 2011 (Special) : 1068-1077. doi: 10.3934/proc.2011.2011.1068

[20]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]