The Second Hessian Type Equation on Almost Hermitian Manifolds

Jianchun Chu , Liding Huang , Xiaohua Zhu

Frontiers of Mathematics ›› : 1 -28.

PDF
Frontiers of Mathematics ›› : 1 -28. DOI: 10.1007/s11464-024-0018-4
Research Article

The Second Hessian Type Equation on Almost Hermitian Manifolds

Author information +
History +
PDF

Abstract

In this paper, we derive the second order estimate to the 2nd Hessian type equation on a compact almost Hermitian manifold.

Keywords

Almost complex manifolds / the 2nd Hessian type equation / Γk(M)-space / the second order estimate

Cite this article

Download citation ▾
Jianchun Chu, Liding Huang, Xiaohua Zhu. The Second Hessian Type Equation on Almost Hermitian Manifolds. Frontiers of Mathematics 1-28 DOI:10.1007/s11464-024-0018-4

登录浏览全文

4963

注册一个新账户 忘记密码

References

[1]

Chu J, Tosatti V, Weinkove B. The Monge–Ampère equation for non-integrable almost complex structures. J. Eur. Math. Soc. (JEMS), 2019, 21(7): 1949-1984.

[2]

Chu J., Huang L., Zhang J., Fully non-linear elliptic equations on compact almost Hermitian manifolds. Calc. Var. Partial Differential Equations, 2023, 62(3): Paper No. 105, 34 pp.

[3]

Chu J, Huang L, Zhu X. The Fu–Yau Equation in higher dimensions. Peking Math. J., 2019, 2(1): 71-97.

[4]

Chu J, Huang L, Zhu X. The Fu–Yau equation on compact astheno-Kähler manifolds. Adv. Math., 2019, 346: 908-945.

[5]

De Bartolomeis P, Tomassini A. On solvable generalized Calabi–Yau manifolds. Ann. Inst. Fourier (Grenoble), 2006, 56(5): 1281-1296.

[6]

Dinew S, Kołodziej S. Liouville and Calabi–Yau type theorems for complex Hessian equations. Amer. J. Math., 2017, 139(2): 403-415.

[7]

Fu J, Yau S-T. A Monge–Ampère-type equation motivated by string theory. Comm. Anal. Geom., 2007, 15(1): 29-75.

[8]

Fu J, Yau S-T. The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation. J. Differential Geom., 2008, 78(3): 369-428.

[9]

Greene BR, Shapere A, Vafa C, Yau S-T. Stringy cosmic strings and noncompact Calabi–Yau manifolds. Nuclear Phys. B, 1990, 337(1): 1-36.

[10]

Harvey FR, Lawson HB. Potential theory on almost complex manifolds. Ann. Inst. Fourier (Grenoble), 2015, 65(1): 171-210.

[11]

Hitchin N. Generalized Calabi–Yau manifolds. Q. J. Math., 2003, 54(3): 281-308.

[12]

Horn RA, Johnson CR. Matrix Analysis, 2013 Second Edition Cambridge: Cambridge University Press

[13]

Hou Z. Complex Hessian equation on Kähler manifold. Int. Math. Res. Not. IMRN, 2009, 16: 3098-3111.

[14]

Hou Z, Ma X, Wu D. A second order estimate for complex Hessian equations on a compact Kähler manifold. Math. Res. Lett., 2010, 17(3): 547-561.

[15]

Huang L., Zhang J., Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds. Math. Z., 2023, 303(2): Paper No. 36, 24 pp.

[16]

Lin M, Trudinger NS. On some inequalities for elementary symmetric functions. Bull. Aust. Math. Soc., 1994, 50: 317-326.

[17]

Phong DH, Picard S, Zhang X. A second order estimate for general complex Hessian equations. Anal. PDE, 2016, 9(7): 1693-1709.

[18]

Phong DH, Picard S, Zhang X. The Fu–Yau equation with negative slope parameter. Invent. Math., 2017, 209(2): 541-576.

[19]

Phong DH, Picard S, Zhang X. On estimates for the Fu–Yau generalization of a Strominger system. J. Reine Angew. Math., 2019, 751: 243-274.

[20]

Phong DH, Picard S, Zhang X. Fu–Yau Hessian equations. J. Differential Geom., 2021, 118(1): 147-187.

[21]

Streets J, Tian G. Generalized Kähler geometry and the pluriclosed flow. Nuclear Phys. B, 2012, 858(2): 366-376.

[22]

Strominger A. Superstrings with torsion. Nuclear Phys. B, 1986, 274(2): 253-284.

[23]

Székelyhidi G. Fully non-linear elliptic equations on compact Hermitian manifolds. J. Differential Geom., 2018, 109(2): 337-378.

[24]

Székelyhidi G, Tosatti V, Weinkove B. Gauduchon metrics with prescribed volume form. Acta Math., 2017, 219(1): 181-211.

[25]

Zhang D. Hessian equations on closed Hermitian manifolds. Pacific J. Math., 2017, 291(2): 485-510.

AI Summary AI Mindmap
PDF

77

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/