June  2021, 20(6): 2361-2377. doi: 10.3934/cpaa.2021085

A class of the non-degenerate complex quotient equations on compact Kähler manifolds

a. 

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

b. 

School of Mathematics and Statistics, Fuyang Normal University, Fuyang 236037, Anhui, China

Received  December 2020 Revised  April 2021 Published  June 2021 Early access  May 2021

Fund Project: Research of the author was supported by the Natural Science Foundation of Anhui Province Education Department (grant nos. KJ2017A341 and KJ2018A0330) and the Talent Project of Fuyang Normal University (grant no. RCXM201714)

In this paper, we are concerned with the equations that are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix on compact K$ \ddot{a} $hler manifolds. Under the assumption of the cone condition, we obtain a priori estimates for the class of complex quotient equations. Then using the method of continuity, we prove an existence result.

Citation: Jundong Zhou. A class of the non-degenerate complex quotient equations on compact Kähler manifolds. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2361-2377. doi: 10.3934/cpaa.2021085
References:
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Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756.   Google Scholar

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P. Cherrier, Equations de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s Hermitienne compactes, Bull. Sc. Math., 2 (1987), 343-385.   Google Scholar

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V. TosattiY. WangB. Weinkove and X. Yang, $C^{2, \alpha}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. Partial Differ. Equ., 54 (2015), 431-453.  doi: 10.1007/s00526-014-0791-0.  Google Scholar

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S. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp$\grave{e}$re equation, I, Commun. Pure Appl. Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.  Google Scholar

[23]

X. Zhang and X. Zhang, Regularity estimates of solutions to complex Monge-Amp$\grave{e}$re equations on Hermitian manifolds, J. Funct. Anal., 260 (2011), 2004-2026.  doi: 10.1016/j.jfa.2010.12.024.  Google Scholar

[24]

D. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.  doi: 10.2140/pjm.2017.291.485.  Google Scholar

show all references

References:
[1]

Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756.   Google Scholar

[2]

P. Cherrier, Equations de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s Hermitienne compactes, Bull. Sc. Math., 2 (1987), 343-385.   Google Scholar

[3]

J. Chu and N. McCleerey, Fully non-linear degenerate elliptic equations in complex geometry, preprint, arXiv: 2010.03431v1. Google Scholar

[4]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, Amer. J. Math., 139 (2017), 403-415.  doi: 10.1353/ajm.2017.0009.  Google Scholar

[5]

H. FangM. Lai and X. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.  doi: 10.1515/CRELLE.2011.027.  Google Scholar

[6]

B. Guan and Q. Li, Complex Monge-Amp$\grave{e}$re equations and totally real submanifolds, Adv. Math., 225 (2010), 1185-1223.  doi: 10.1016/j.aim.2010.03.019.  Google Scholar

[7]

P. Guan and X. Zhang, A class of curvature type equations. Pure and Applied Math Quarterly, preprint, arXiv: 1909.03645. Google Scholar

[8]

A. Hanani, Equations du type de Monge-Amp$\grave{e}$re sur les vari$\acute{e}$t$\acute{e}$s hermitiennes compactes, J. Funct. Anal., 137 (1996), 49-75.  doi: 10.1006/jfan.1996.0040.  Google Scholar

[9]

Z. L. HouX. N. Ma and D. M. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.  doi: 10.4310/MRL.2010.v17.n3.a12.  Google Scholar

[10]

N. V. Krylov, On the general notion of fully nonlinear second order elliptic equation, Trans. Amer. Math. Soc., 3 (1995), 857-895.  doi: 10.2307/2154876.  Google Scholar

[11]

M. Lin and N. S. Trudinger., On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50 (1994), 317-326.  doi: 10.1017/S0004972700013770.  Google Scholar

[12]

C. LiC. Ren and Z. Wang, Curvature estimates for convex solutions of some fully nonlinear Hessian-type equations, Calc. Var. Partial Differ. Equ., 58 (2019), 1-32.  doi: 10.1007/s00526-019-1623-z.  Google Scholar

[13]

D. Phong and T. Dat, Fully non-linear parabolic equations on compact Hermitian manifolds, preprint, arXiv: 1711.10697v2. Google Scholar

[14]

J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309.  Google Scholar

[15]

J. Song and B. Weinkove, On the convergence and singularities of the J -flow with applications to the Mabuchi energy, Commun. Pure Appl. Math., 61 (2008), 210-229.  doi: 10.1002/cpa.20182.  Google Scholar

[16]

W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, J. Geom. Anal., 26 (2016), 2459-2473.  doi: 10.1007/s12220-015-9634-2.  Google Scholar

[17]

W. Sun, On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: $L^{\infty}$estimate, Commun. Pure Appl. Math., 70 (2017), 172-199.  doi: 10.1002/cpa.21652.  Google Scholar

[18]

G. Sz$\acute{e}$kelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, J. Differ. Geom., 109 (2018), 337-378.  doi: 10.4310/jdg/1527040875.  Google Scholar

[19]

V. Tosatti and B. Weinkove, Estimates for the complex Monge-Amp$\grave{e}$re equation on Hermitian and balanced manifolds, Asian J. Math., 14 (2010), 19-40.  doi: 10.4310/AJM.2010.v14.n1.a3.  Google Scholar

[20]

V. Tosatti and B. Weinkove, The complex Monge-Amp$\grave{e}$re equation on compact Hermitian manifolds, J. Amer. Math. Soc., 23 (2010), 1187-1195.  doi: 10.1090/S0894-0347-2010-00673-X.  Google Scholar

[21]

V. TosattiY. WangB. Weinkove and X. Yang, $C^{2, \alpha}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. Partial Differ. Equ., 54 (2015), 431-453.  doi: 10.1007/s00526-014-0791-0.  Google Scholar

[22]

S. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp$\grave{e}$re equation, I, Commun. Pure Appl. Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.  Google Scholar

[23]

X. Zhang and X. Zhang, Regularity estimates of solutions to complex Monge-Amp$\grave{e}$re equations on Hermitian manifolds, J. Funct. Anal., 260 (2011), 2004-2026.  doi: 10.1016/j.jfa.2010.12.024.  Google Scholar

[24]

D. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510.  doi: 10.2140/pjm.2017.291.485.  Google Scholar

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