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

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

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

Jundong Zhou ^a,b,

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.

Keywords: Complex quotient equations, Kähler manifolds, cone condition.

Mathematics Subject Classification: Primary: 35J60; Secondary: 35B45.

Citation: Jundong Zhou. A class of the non-degenerate complex quotient equations on compact Kähler manifolds. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2361-2377. doi: 10.3934/cpaa.2021085

References:

[1]	Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756.
[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.
[3]	J. Chu and N. McCleerey, Fully non-linear degenerate elliptic equations in complex geometry, preprint, arXiv: 2010.03431v1.
[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.
[5]	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. doi: 10.1515/CRELLE.2011.027.
[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.
[7]	P. Guan and X. Zhang, A class of curvature type equations. Pure and Applied Math Quarterly, preprint, arXiv: 1909.03645.
[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.
[9]	Z. L. Hou, X. 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.
[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.
[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.
[12]	C. Li, C. 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.
[13]	D. Phong and T. Dat, Fully non-linear parabolic equations on compact Hermitian manifolds, preprint, arXiv: 1711.10697v2.
[14]	J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	V. Tosatti, Y. Wang, B. 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.
[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.
[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.
[24]	D. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510. doi: 10.2140/pjm.2017.291.485.

show all references

References:

[1]	Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55 (2005), 1735-1756.
[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.
[3]	J. Chu and N. McCleerey, Fully non-linear degenerate elliptic equations in complex geometry, preprint, arXiv: 2010.03431v1.
[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.
[5]	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. doi: 10.1515/CRELLE.2011.027.
[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.
[7]	P. Guan and X. Zhang, A class of curvature type equations. Pure and Applied Math Quarterly, preprint, arXiv: 1909.03645.
[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.
[9]	Z. L. Hou, X. 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.
[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.
[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.
[12]	C. Li, C. 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.
[13]	D. Phong and T. Dat, Fully non-linear parabolic equations on compact Hermitian manifolds, preprint, arXiv: 1711.10697v2.
[14]	J. Spruck, Geometric aspects of the theory of fully nonlinear elliptic equations, Clay Mathematics Proceedings, 2 (2005), 283-309.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	V. Tosatti, Y. Wang, B. 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.
[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.
[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.
[24]	D. Zhang, Hessian equations on closed Hermitian manifolds, Pacific J. Math., 291 (2017), 485-510. doi: 10.2140/pjm.2017.291.485.

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