American Institute of Mathematical Sciences

Advanced
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 & 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.   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. 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.  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. 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.  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. 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.  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. 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.  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

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. 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.  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. 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.  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. 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.  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. 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.  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

[1]

Dmitry Jakobson, Alexander Strohmaier, Steve Zelditch. On the spectrum of geometric operators on Kähler manifolds. Journal of Modern Dynamics, 2008, 2 (4) : 701-718. doi: 10.3934/jmd.2008.2.701
[2]

Carlos Kenig, Tobias Lamm, Daniel Pollack, Gigliola Staffilani, Tatiana Toro. The Cauchy problem for Schrödinger flows into Kähler manifolds. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 389-439. doi: 10.3934/dcds.2010.27.389
[3]

Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137
[4]

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
[5]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377
[6]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[7]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641
[8]

M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3429-3463. doi: 10.3934/cpaa.2021113
[9]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272
[10]

Xianmin Geng, Shengli Zhou, Jiashan Tang, Cong Yang. A sufficient condition for classified networks to possess complex network features. Networks & Heterogeneous Media, 2012, 7 (1) : 59-69. doi: 10.3934/nhm.2012.7.59
[11]

Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2139-2154. doi: 10.3934/cpaa.2021061
[12]

Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021126
[13]

D.J. Georgiev, A. J. Roberts, D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model. Conference Publications, 2007, 2007 (Special) : 419-428. doi: 10.3934/proc.2007.2007.419
[14]

Yazhou Han. Integral equations on compact CR manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2187-2204. doi: 10.3934/dcds.2020358
[15]

Tiancong Chen, Qing Han. Smooth local solutions to Weingarten equations and $\sigma_k$-equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 653-660. doi: 10.3934/dcds.2016.36.653
[16]

Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641
[17]

Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic & Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001
[18]

Zhihua Huang, Xiaochun Liu. Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3201-3216. doi: 10.3934/cpaa.2019144
[19]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[20]

Luis Barreira, Claudia Valls. Stable manifolds with optimal regularity for difference equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1537-1555. doi: 10.3934/dcds.2012.32.1537

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (93)
  • HTML views (128)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences