A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD

doi:10.1007/s10473-022-0518-9

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 2040-2062.doi: 10.1007/s10473-022-0518-9

• 论文 • 上一篇

A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD

Jiaogen ZHANG

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

收稿日期:2020-08-19 修回日期:2022-04-01 发布日期:2022-11-02
通讯作者: Jiaogen Zhang,E-mail:zjgmath@ustc.edu.cn E-mail:zjgmath@ustc.edu.cn
基金资助:
The research was supported by the National Key R and D Program of China (2020YFA0713100).

A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD

Jiaogen ZHANG

School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

Received:2020-08-19 Revised:2022-04-01 Published:2022-11-02
Contact: Jiaogen Zhang,E-mail:zjgmath@ustc.edu.cn E-mail:zjgmath@ustc.edu.cn
Supported by:
The research was supported by the National Key R and D Program of China (2020YFA0713100).

摘要/Abstract

摘要： Let Ω ⊆M be a bounded domain with a smooth boundary ∂Ω, where (M, J, g) is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on Ω. Under the existence of a C²-smooth strictly J-plurisubharmonic (J-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.

关键词: complex Monge-Ampère equation, almost Hermitian manifold, a priori estimate, subsolution, J-plurisubharmonic

Abstract: Let Ω ⊆M be a bounded domain with a smooth boundary ∂Ω, where (M, J, g) is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on Ω. Under the existence of a C²-smooth strictly J-plurisubharmonic (J-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.

Key words: complex Monge-Ampère equation, almost Hermitian manifold, a priori estimate, subsolution, J-plurisubharmonic

中图分类号:

32W20

Jiaogen ZHANG. A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD[J]. 数学物理学报(英文版), 2022, 42(5): 2040-2062.

Jiaogen ZHANG. A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD[J]. Acta mathematica scientia,Series B, 2022, 42(5): 2040-2062.

参考文献

[1] Akramov I, Oliver M. On the existence of solutions to Bi-planar Monge-Ampére equation. Acta Math Sci, 2020, 40B(2): 379–388
[2] Bedford E, Taylor B A. The Dirichlet problem for a complex Monge-Ampère equation. Invent Math, 1976, 37(1): 1–44
[3] Bedford E, Taylor B A. Variational properties of the complex Monge-Ampère equation. I. Dirichlet principle. Duke Math J, 1978, 45(2): 375–403
[4] Błocki Z. On geodesics in the space of Kähler metrics. Preprint available on the website of the author 2009
[5] Caffarelli L, Kohn J J, Nirenberg L, Spruck J. The Dirichlet problem for nonlinear second-order elliptic equations II. Complex Monge-Ampère, and uniformly elliptic equations. Comm Pure Appl Math, 1985, 38(2): 209–252
[6] Caffarelli L, Nirenberg L, Spruck J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math, 1985, 155(3/4): 261–301
[7] Chen X. The space of Kähler metrics. J Differential Geom, 2000, 56(2): 189–234
[8] Chu J, Tosatti V, Weinkove B. The Monge-Ampère equation for non-integrable almost complex structures. J Eur Math Soc, 2019, 21(7): 1949–1984
[9] Ehresmann C, Libermann P. Sur les structures presque hermitiennes isotropes. C R Acad Sci Paris, 1951, 232: 1281–1283
[10] Feng K, Shi Y, Xu Y. On the Dirichlet problem for a class of singular complex Monge-Ampère equations. Acta Math Sin Engl Ser, 2018, 34(2): 209–220
[11] Székelyhidi G. Fully nonlinear elliptic equations on compact Hermitian manifolds. J Differential Geom, 2018, 109(2): 337–378
[12] Gauduchon P. Hermitian connections and Dirac operators. Boll Un Mat Ital, 1997, 11B(suppl): 257–288
[13] Guan B. The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function. Comm Anal Geom, 1998, 6(4): 687–703
[14] Guan B. The Dirichlet problem for complex Monge-Ampère equations and applications//Trends in Partial Differential Equations. Adv Lect Math, 10. Somerville, MA: Int Press, 2010: 53–97
[15] Guan B. Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math J, 2014, 163(8): 1491–1524
[16] Guan P. The extremal function associated to intrinsic norms. Ann of Math, 2002, 156(1): 197–211
[17] Harvey F R, Lawson B. Potential theory on almost complex manifolds. Ann Inst Fourier (Grenoble), 2015, 65(1): 171–210
[18] He W. On the regularity of the complex Monge-Ampère equations. Proc Amer Math Soc, 2012, 140(5): 1719–1727
[19] Hwang S. Cauchy’s interlace theorem for eigenvalues of Hermitian matrices. Am Math Mon, 2004, 111(2): 157–159
[20] Jiang F, Yang X. Weak solutions of Monge-Ampére equation in optimal transportation. Acta Math Sci, 2013, 33B(4): 950–962
[21] Kobayashi S, Nomizu K. Foundations of Differential Geometry, vol I. New York: Interscience Publishers, Wiley, 1963
[22] Li C, Li J, Zhang X. A mean value formula and a Liouville theorem for the complex Monge-Ampère equation. Int Math Res Not, 2020, 3: 853–867
[23] Li C, Li J, Zhang X. Some interior regularity estimates for solutions of complex Monge-Ampère equations on a ball. Calc Var Partial Differential Equations, 2021, 60 (1): art. 34
[24] Li S. On the existence and regularity of Dirichlet problem for complex Monge-Ampère equations on weakly pseudoconvex domains. Calc Var Partial Differential Equations, 2004, 20(2): 119–132
[25] Li C, Zheng T. The Dirichlet problem on almost Hermitian manifolds. J Geom Anal, 2021, 31(6): 6452–6480
[26] Pali N. Fonctions plurisousharmoniques et courants positifs de type (1, 1) sur une variété presque complexe. Manuscripta Math, 2005, 118(3): 311–337
[27] Pliś S. The Monge-Ampère equation on almost Hermitian manifolds. Math Z, 2014, 276: 969–983
[28] Schulz F. A C² estimate for solutions of complex Monge-Ampère equations. J Reine Angew Math, 1984, 348: 88–93
[29] Spruck J. Geometric aspects of the theory of fully nonlinear elliptic equations//Global Theory of Minimal Surfaces, vol 2. Providence, RI: Amer Math Soc, 2005: 283–309
[30] Trudinger N S. On the Dirichlet problem for Hessian equations. Acta Math, 1995, 175(2): 151–164
[31] Tosatti V. A general Schwarz lemma for almost Hermitian manifolds. Comm Anal Geom, 2007, 15(5): 1063–1086
[32] Wang Y, Weinkove B, Yang X. C^2,α estimates for nonlinear elliptic equations in complex and almost complex geometry. Calc Var Partial Differential Equations, 2015, 54(1): 431–453
[33] Tosatti V, Weinkove B, Yau S T. Taming symplectic forms and the Calabi-Yau equation. Proc Lond Math Soc, 2008, 97(2): 401–424
[34] Wang Y. On the C^2,α-regularity of the complex Monge-Ampère equation. Math Res Lett, 2012, 19(4): 939–946
[35] Wang Y, Zhang X. Dirichlet problem for Hermitian-Einstein equation over almost Hermitian manifold. Acta Math Sin Engl Ser, 2012, 28(6): 1249–1260
[36] Zhang X. Twisted quiver bundles over almost complex manifolds. J Geom Phys, 2005, 55(3): 267–290

A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD

A SUBSOLUTION THEOREM FOR THE MONGE-AMPÈRE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD

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