November  2017, 37(11): 5707-5730. doi: 10.3934/dcds.2017247

On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received  May 2017 Published  July 2017

Fund Project: Research supported in part by NSF in China, No. 11571288, No. 11671330, No. 11571332, No. 11625106 and No. 11131007.

In this paper, we study a class of fully nonlinear elliptic equations on annuli of metric cones constructed from closed Sasakian manifolds and derive the a priori estimates assuming the existence of subsolutions. Moreover, such a priori estimates can be applied to certain degenerate equations. A condition for the solvability of Dirichlet problem for non-degenerate fully nonlinear elliptic equations is discovered. Furthermore, we also discuss degenerate equations.

Citation: Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
References:
[1]

T. Aubin, Équations du type Monge-Ampère sur les variétés Kähleriennes compactes, (French), Bull. Sci. Math., 102 (1978), 63-95.   Google Scholar

[2]

E. Bedford and B. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37 (1976), 1-4.  doi: 10.1007/BF01418826.  Google Scholar

[3]

Z. Blocki, On geodesics in the space of Kähler metrics, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 37 (1976), 1-44.   Google Scholar

[4]

C. Boyer and K. Galicki, Sasakian Geometry, Oxford: Oxford Mathematical Monographs, Oxford University press, 2008.  Google Scholar

[5]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ: Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[6]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Rev. Mat. Iberoamericana, 2 (1986), 19-27.  doi: 10.4171/RMI/23.  Google Scholar

[7]

L. CaffarelliJ. KohnL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅱ. Complex Monge-Ampère, and uniformaly elliptic, equations, Comm. Pure Applied Math., 38 (1985), 209-252.  doi: 10.1002/cpa.3160380206.  Google Scholar

[8]

E. Calabi, The Space of Kähler Metrics, Proc. Internat. Congress of Mathematicians, Amsterdam, Holland. 1954. Google Scholar

[9]

X.-X. Chen, The space of Kähler metrics, J. Diff. Geom., 56 (2000), 189-234.  doi: 10.4310/jdg/1090347643.  Google Scholar

[10]

X.-X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852.  doi: 10.4310/CAG.2004.v12.n4.a4.  Google Scholar

[11]

T. Collins, A. Jacob and S. -T. Yau, $(1, 1)$ forms with special Lagrangian type: A priori estimates and algebraic obstructions, arXiv: 1508.01934. Google Scholar

[12]

S.-K. Donaldson, Symmeric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, American Mathematical Society Translations: Series 2, 196 . Providence, RI: American Mmathematical Society, 45 (1999), 13-33.  doi: 10.1090/trans2/196/02.  Google Scholar

[13]

S.-K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 45 (1999), 13-33.  doi: 10.4310/AJM.1999.v3.n1.a1.  Google Scholar

[14]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, American Journal of Mathematics, 139 (2017), 403-415, arXiv: 1203.3995. doi: 10.1353/ajm.2017.0009.  Google Scholar

[15]

L. Evans, Classical solutions of fully nonlinear convex, second order elliptic equations, Comm. Pure Applied Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

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H. FangM.-J. Lai and X.-N. 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

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J.-X. Fu, On non-Kähler Calabi-Yau threefolds with Balanced metrics, Proc. Internat. Congress of Mathematicians, Hyderabad, India. New Delhi: Hindustan Book Agency, (2010), 705-716.   Google Scholar

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A. FutakiH. Ono and G.-F. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom., 83 (2009), 585-635.  doi: 10.4310/jdg/1264601036.  Google Scholar

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P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518.  doi: 10.1007/BF01455968.  Google Scholar

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J. GauntlettD. MartelliJ. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys., 8 (2004), 987-1000.  doi: 10.4310/ATMP.2004.v8.n6.a3.  Google Scholar

[21]

M. GodlinskiW. Kopczynski and P. Nurowski, Locally Sasakian manifolds, Classical quantum gravity, 17 (2000), 105-115.  doi: 10.1088/0264-9381/17/18/101.  Google Scholar

[22]

B. Guan, The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom., 6 (1998), 687-703.  doi: 10.4310/CAG.1998.v6.n4.a3.  Google Scholar

[23]

B. Guan, Second order estimates and regularity for fully nonlinear ellitpic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.  Google Scholar

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B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, preprint. Google Scholar

[25]

B. Guan and Q. Li, The Dirichlet problem for a complex Monge-Ampère type equation on Hermitian manifolds, Adv. Math., 246 (2013), 351-367.  doi: 10.1016/j.aim.2013.07.006.  Google Scholar

[26]

B. Guan and X. -L. Nie, Fully nonlinear elliptic equations on Hermitian manifolds, preprint. Google Scholar

[27]

B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. PDE., 54 (2013), 901-916.  doi: 10.1007/s00526-014-0810-1.  Google Scholar

[28]

B. GuanS.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE., 8 (2015), 1145-1164.  doi: 10.2140/apde.2015.8.1145.  Google Scholar

[29]

B. Guan and J. Spruck, Boundary-value problems on $\mathbb{S}^{n}$ for surfaces of constant Gauss curvature, Ann. Math., 138 (1993), 601-624.  doi: 10.2307/2946558.  Google Scholar

[30]

P.-F. Guan, The extremal function associated to intrinsic norms, Ann. Math., 156 (2002), 197-211.  doi: 10.2307/3597188.  Google Scholar

[31]

P.-F. GuanN. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104.  doi: 10.1007/BF02392824.  Google Scholar

[32]

P.-F. Guan and X. Zhang, A geodesic equation in the space of Sasake metrics, Geometry and Analysis, Adv. Lect. Math., Somerville: International Press, 17 (2011), 303-318.   Google Scholar

[33]

P.-F. Guan and X. Zhang, Regularity of the geodesic equation in the space of Sasake metrics, Adv. Math., 230 (2012), 321-371.  doi: 10.1016/j.aim.2011.12.002.  Google Scholar

[34]

D. HoffmanH. Rosenberg and J. Spruck, Boundary value problems for surfaces of constant Gauss Curvature, Comm. Pure Applied Math., 45 (1992), 1051-1062.  doi: 10.1002/cpa.3160450807.  Google Scholar

[35]

Z. 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

[36]

N. Ivochkina, The integral method of barrier functions and the Dirichlet problem for equations with operators of the Monge-Ampère type, (Russian)Mat. Sb. (N.S.), 112 (1980), 193-206; English transl.: Math. USSR Sb., 40 (1981), 179-192.  Google Scholar

[37]

N. IvochkinaN. Trudinger and X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. PDE., 29 (2004), 219-235.  doi: 10.1081/PDE-120028851.  Google Scholar

[38]

N. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.   Google Scholar

[39]

Y.-Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Applied Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[40]

Y.-Y. Li, Degenerate conformally invariant fully nonlinear elliptic equations, Arch. Ration. Mech. Anal., 186 (2007), 25-51.  doi: 10.1007/s00205-006-0041-5.  Google Scholar

[41]

T. Mabuchi, Some symplectic geometry on Kähler manifolds. I, Osaka J. Math., 24 (1987), 227-252.   Google Scholar

[42]

D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys., 262 (2006), 51-89.  doi: 10.1007/s00220-005-1425-3.  Google Scholar

[43]

D. MartelliJ. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Mathe. Phys., 280 (2008), 611-673.  doi: 10.1007/s00220-008-0479-4.  Google Scholar

[44]

D. -H. Phong, S. Picard and X. -W. Zhang, On estimates for the Fu-Yau generalization of a Strominger system, arXiv: 1507.08193. doi: 10.1515/crelle-2016-0052.  Google Scholar

[45]

D.-H. Phong and J. Sturm, The Dirichlet problem for degenerate complex Monge-Ampère equations, Comm. Anal. Geom., 18 (2010), 145-170.  doi: 10.4310/CAG.2010.v18.n1.a6.  Google Scholar

[46]

D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bulletin de la SMF, 143 (2015), 763-800, arXiv: 1310.3685. doi: 10.24033/bsmf.2704.  Google Scholar

[47]

S. Semmes, Complex Monge-Ampère and sympletic manifolds, Amer. J. Math., 114 (1992), 495-550.  doi: 10.2307/2374768.  Google Scholar

[48]

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

[49]

W. Sun, Generalized complex Monge-Ampère type equations on closed Hermitian manifolds, arXiv: 1412.8192. Google Scholar

[50]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, arXiv: 1501.02762. Google Scholar

[51]

G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, arXiv: 1503.04491. Google Scholar

[52]

M. E. Taylor, Partial Differential Equations I, Basic Theory, New York, Berlin, Heidelberg: Applied Mathematical Sciences, 115, Springer-Verlag, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[53]

G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc., 3 (1990), 579-609.  doi: 10.2307/1990928.  Google Scholar

[54]

V. Tosatti and B. Weinkove, Hermitian metrics, $(n-1, n-1)$-forms and Monge-Ampère equations, arXiv: 1310.6326. Google Scholar

[55]

N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.  Google Scholar

[56]

S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Applied Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.  Google Scholar

show all references

References:
[1]

T. Aubin, Équations du type Monge-Ampère sur les variétés Kähleriennes compactes, (French), Bull. Sci. Math., 102 (1978), 63-95.   Google Scholar

[2]

E. Bedford and B. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37 (1976), 1-4.  doi: 10.1007/BF01418826.  Google Scholar

[3]

Z. Blocki, On geodesics in the space of Kähler metrics, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 37 (1976), 1-44.   Google Scholar

[4]

C. Boyer and K. Galicki, Sasakian Geometry, Oxford: Oxford Mathematical Monographs, Oxford University press, 2008.  Google Scholar

[5]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ: Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.  Google Scholar

[6]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Rev. Mat. Iberoamericana, 2 (1986), 19-27.  doi: 10.4171/RMI/23.  Google Scholar

[7]

L. CaffarelliJ. KohnL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅱ. Complex Monge-Ampère, and uniformaly elliptic, equations, Comm. Pure Applied Math., 38 (1985), 209-252.  doi: 10.1002/cpa.3160380206.  Google Scholar

[8]

E. Calabi, The Space of Kähler Metrics, Proc. Internat. Congress of Mathematicians, Amsterdam, Holland. 1954. Google Scholar

[9]

X.-X. Chen, The space of Kähler metrics, J. Diff. Geom., 56 (2000), 189-234.  doi: 10.4310/jdg/1090347643.  Google Scholar

[10]

X.-X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852.  doi: 10.4310/CAG.2004.v12.n4.a4.  Google Scholar

[11]

T. Collins, A. Jacob and S. -T. Yau, $(1, 1)$ forms with special Lagrangian type: A priori estimates and algebraic obstructions, arXiv: 1508.01934. Google Scholar

[12]

S.-K. Donaldson, Symmeric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, American Mathematical Society Translations: Series 2, 196 . Providence, RI: American Mmathematical Society, 45 (1999), 13-33.  doi: 10.1090/trans2/196/02.  Google Scholar

[13]

S.-K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 45 (1999), 13-33.  doi: 10.4310/AJM.1999.v3.n1.a1.  Google Scholar

[14]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, American Journal of Mathematics, 139 (2017), 403-415, arXiv: 1203.3995. doi: 10.1353/ajm.2017.0009.  Google Scholar

[15]

L. Evans, Classical solutions of fully nonlinear convex, second order elliptic equations, Comm. Pure Applied Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.  Google Scholar

[16]

H. FangM.-J. Lai and X.-N. 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

[17]

J.-X. Fu, On non-Kähler Calabi-Yau threefolds with Balanced metrics, Proc. Internat. Congress of Mathematicians, Hyderabad, India. New Delhi: Hindustan Book Agency, (2010), 705-716.   Google Scholar

[18]

A. FutakiH. Ono and G.-F. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom., 83 (2009), 585-635.  doi: 10.4310/jdg/1264601036.  Google Scholar

[19]

P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518.  doi: 10.1007/BF01455968.  Google Scholar

[20]

J. GauntlettD. MartelliJ. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys., 8 (2004), 987-1000.  doi: 10.4310/ATMP.2004.v8.n6.a3.  Google Scholar

[21]

M. GodlinskiW. Kopczynski and P. Nurowski, Locally Sasakian manifolds, Classical quantum gravity, 17 (2000), 105-115.  doi: 10.1088/0264-9381/17/18/101.  Google Scholar

[22]

B. Guan, The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom., 6 (1998), 687-703.  doi: 10.4310/CAG.1998.v6.n4.a3.  Google Scholar

[23]

B. Guan, Second order estimates and regularity for fully nonlinear ellitpic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.  Google Scholar

[24]

B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, preprint. Google Scholar

[25]

B. Guan and Q. Li, The Dirichlet problem for a complex Monge-Ampère type equation on Hermitian manifolds, Adv. Math., 246 (2013), 351-367.  doi: 10.1016/j.aim.2013.07.006.  Google Scholar

[26]

B. Guan and X. -L. Nie, Fully nonlinear elliptic equations on Hermitian manifolds, preprint. Google Scholar

[27]

B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. PDE., 54 (2013), 901-916.  doi: 10.1007/s00526-014-0810-1.  Google Scholar

[28]

B. GuanS.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE., 8 (2015), 1145-1164.  doi: 10.2140/apde.2015.8.1145.  Google Scholar

[29]

B. Guan and J. Spruck, Boundary-value problems on $\mathbb{S}^{n}$ for surfaces of constant Gauss curvature, Ann. Math., 138 (1993), 601-624.  doi: 10.2307/2946558.  Google Scholar

[30]

P.-F. Guan, The extremal function associated to intrinsic norms, Ann. Math., 156 (2002), 197-211.  doi: 10.2307/3597188.  Google Scholar

[31]

P.-F. GuanN. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104.  doi: 10.1007/BF02392824.  Google Scholar

[32]

P.-F. Guan and X. Zhang, A geodesic equation in the space of Sasake metrics, Geometry and Analysis, Adv. Lect. Math., Somerville: International Press, 17 (2011), 303-318.   Google Scholar

[33]

P.-F. Guan and X. Zhang, Regularity of the geodesic equation in the space of Sasake metrics, Adv. Math., 230 (2012), 321-371.  doi: 10.1016/j.aim.2011.12.002.  Google Scholar

[34]

D. HoffmanH. Rosenberg and J. Spruck, Boundary value problems for surfaces of constant Gauss Curvature, Comm. Pure Applied Math., 45 (1992), 1051-1062.  doi: 10.1002/cpa.3160450807.  Google Scholar

[35]

Z. 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

[36]

N. Ivochkina, The integral method of barrier functions and the Dirichlet problem for equations with operators of the Monge-Ampère type, (Russian)Mat. Sb. (N.S.), 112 (1980), 193-206; English transl.: Math. USSR Sb., 40 (1981), 179-192.  Google Scholar

[37]

N. IvochkinaN. Trudinger and X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. PDE., 29 (2004), 219-235.  doi: 10.1081/PDE-120028851.  Google Scholar

[38]

N. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108.   Google Scholar

[39]

Y.-Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Applied Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.  Google Scholar

[40]

Y.-Y. Li, Degenerate conformally invariant fully nonlinear elliptic equations, Arch. Ration. Mech. Anal., 186 (2007), 25-51.  doi: 10.1007/s00205-006-0041-5.  Google Scholar

[41]

T. Mabuchi, Some symplectic geometry on Kähler manifolds. I, Osaka J. Math., 24 (1987), 227-252.   Google Scholar

[42]

D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys., 262 (2006), 51-89.  doi: 10.1007/s00220-005-1425-3.  Google Scholar

[43]

D. MartelliJ. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Mathe. Phys., 280 (2008), 611-673.  doi: 10.1007/s00220-008-0479-4.  Google Scholar

[44]

D. -H. Phong, S. Picard and X. -W. Zhang, On estimates for the Fu-Yau generalization of a Strominger system, arXiv: 1507.08193. doi: 10.1515/crelle-2016-0052.  Google Scholar

[45]

D.-H. Phong and J. Sturm, The Dirichlet problem for degenerate complex Monge-Ampère equations, Comm. Anal. Geom., 18 (2010), 145-170.  doi: 10.4310/CAG.2010.v18.n1.a6.  Google Scholar

[46]

D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bulletin de la SMF, 143 (2015), 763-800, arXiv: 1310.3685. doi: 10.24033/bsmf.2704.  Google Scholar

[47]

S. Semmes, Complex Monge-Ampère and sympletic manifolds, Amer. J. Math., 114 (1992), 495-550.  doi: 10.2307/2374768.  Google Scholar

[48]

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

[49]

W. Sun, Generalized complex Monge-Ampère type equations on closed Hermitian manifolds, arXiv: 1412.8192. Google Scholar

[50]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, arXiv: 1501.02762. Google Scholar

[51]

G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, arXiv: 1503.04491. Google Scholar

[52]

M. E. Taylor, Partial Differential Equations I, Basic Theory, New York, Berlin, Heidelberg: Applied Mathematical Sciences, 115, Springer-Verlag, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[53]

G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc., 3 (1990), 579-609.  doi: 10.2307/1990928.  Google Scholar

[54]

V. Tosatti and B. Weinkove, Hermitian metrics, $(n-1, n-1)$-forms and Monge-Ampère equations, arXiv: 1310.6326. Google Scholar

[55]

N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.  Google Scholar

[56]

S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Applied Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.  Google Scholar

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