On \(L^\infty\) estimates for complex Monge-Ampère equations. (English) Zbl 1525.35121

\(L^{\infty} \) estimates for the complex Monge-Ampère equations on compact Kähler manifolds, via Moser iteration, were one of the crucial points in Yau’s proof of the Calabi conjecture. Later the reviewer proposed a different method, based on properties of positive currents, which works for nonnegative right hand side with density in \(L^p\) , \(p>1\). In the present paper, the authors introduce a new approach, using PDEs techniques, which allows to prove the latter result and which is applicable in more general settings.
Let \((X, \omega _X )\) be a compact \(n\)-dimensional Kähler manifold. For a smooth function \(u\) the notation \(\omega _u =\omega _X +i\partial\bar{\partial} u \) is used. Fix local coordinates and compose the matrix of coefficients of \(\omega _u\) with the inverse matrix of \(\omega _X\) to define an endomorphism \(h_u\). Denoting by \(\lambda [h_u]\) the vector of eigenvalues of that endomorphism we consider the equation \[ f ( \lambda [h_u ]) =ce^F, \tag{\(\ast\)}\] where \(c>0\) is a constant (depending on the equation), \(F\) a suitably normalized function, and \(\lambda [h_u] \in \Gamma\) with \(\Gamma\) a cone with vertex at \(0\), contained in the set where the sum of \(\lambda _j\) is positive and containing the first (positive) octant. The operator \(f\) is elliptic (increasing in each \(\lambda _j\)), homogeneous of order \(1\), and it satisfies for some positive \(\gamma\) \[ det \left( \frac{ \partial f ( \lambda [h])}{\partial h_{jk} }\right) >\gamma . \] The given function \(F\) is assumed to have finite \(p\)-entropy \[ \int _X e^{nF}|F|^p \omega _X ^n <\infty \] for \(p>n\). The main theorem says that with the above hypothesis any \(C^2\) solution \(u\) of equation \((*)\) satisfies \[ \sup |u| \leq C , \] where \(C\) depends on \(n,p,\gamma, c, \omega _X \), the volume of \(X\) and the \(p\)-entropy.
The theorem covers the complex Monge-Ampère equation, the complex Hessian equation and the complex quotient Hessian equation. In a second statement it is extended so that the estimete holds (uniformly) for families of metrics approximating a semipositive form, which is no longer Kähler.
The new idea of the proof is the use of the solution \(v\) of an auxiliary complex Monge-Ampère equation with the right hand depending on \(u\) which solves \((*)\). Then the barrier function involving \(u\) and \(v\) is cleverly chosen and it is shown that under the assumptions it must be negative. This finally leads, via a De Giorgi-type lemma, to the estimate on \(u\).


35J60 Nonlinear elliptic equations
35J96 Monge-Ampère equations
32W20 Complex Monge-Ampère operators
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C56 Other complex differential geometry
Full Text: DOI arXiv


[1] Trudinger, Neil S.; Wang, Xu-Jia, Hessian measures. {II}, Ann. of Math. (2). Annals of Mathematics. Second Series, 150, 579-604 (1999) · Zbl 0947.35055 · doi:10.2307/121089
[2] Caffarelli, L.; Nirenberg, L.; Spruck, J., The {D}irichlet problem for nonlinear second-order elliptic equations. {III}. {F}unctions of the eigenvalues of the {H}essian, Acta Math.. Acta Mathematica, 155, 261-301 (1985) · Zbl 0654.35031 · doi:10.1007/BF02392544
[3] Berman, Robert J.; Berndtsson, Bo, Moser-{T}rudinger type inequalities for complex {M}onge-{A}mp\`{e}re operators and {A}ubin’s “hypoth\`{e}se fondamentale”, Ann. Fac. Sci. Toulouse Math. (6). Annales de la Facult\'{e} des Sciences de Toulouse. Math\'{e}matiques. S\'{e}rie 6, 31, 595-645 (2022) · Zbl 07712230 · doi:10.5802/afst.1704
[4] Bedford, Eric; Taylor, B. A., The {D}irichlet problem for a complex {M}onge-{A}mp\`{e}re equation, Invent. Math.. Inventiones Mathematicae, 37, 1-44 (1976) · Zbl 0315.31007 · doi:10.1007/BF01418826
[5] Bedford, Eric; Taylor, B. A., A new capacity for plurisubharmonic functions, Acta Math.. Acta Mathematica, 149, 1-40 (1982) · Zbl 0547.32012 · doi:10.1007/BF02392348
[6] Blocki, Zbigniew, On the uniform estimate in the {C}alabi-{Y}au theorem, {II}, Sci. China Math.. Science China. Mathematics, 54, 1375-1377 (2011) · Zbl 1239.32032 · doi:10.1007/s11425-011-4197-6
[7] Chen, Xiuxiong; Cheng, Jingrui, On the constant scalar curvature {K}\"{a}hler metrics ({I})—{A} priori estimates, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 34, 909-936 (2021) · Zbl 1472.14042 · doi:10.1090/jams/967
[8] Collins, Tristan C.; Sz\'{e}kelyhidi, G\'{a}bor, Convergence of the {\(J\)}-flow on toric manifolds, J. Differential Geom.. Journal of Differential Geometry, 107, 47-81 (2017) · Zbl 1392.53072 · doi:10.4310/jdg/1505268029
[9] Demailly, Jean-Pierre; Pali, Nefton, Degenerate complex {M}onge-{A}mp\`{e}re equations over compact {K}\"{a}hler manifolds, Internat. J. Math.. International Journal of Mathematics, 21, 357-405 (2010) · Zbl 1191.53029 · doi:10.1142/S0129167X10006070
[10] Di Nezza, Eleonora; Guedj, Vincent; Lu, Chinh H., Finite entropy vs finite energy, Comment. Math. Helv.. Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society, 96, 389-419 (2021) · Zbl 1473.32014 · doi:10.4171/cmh/515
[11] Dinew, S.; Kolodziej, S., A priori estimates for complex {H}essian equations, Anal. PDE. Analysis & PDE, 7, 227-244 (2014) · Zbl 1297.32020 · doi:10.2140/apde.2014.7.227
[12] Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed, Singular {K}\"{a}hler-{E}instein metrics, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 22, 607-639 (2009) · Zbl 1215.32017 · doi:10.1090/S0894-0347-09-00629-8
[13] H\"{o}rmander, Lars, An Introduction to Complex Analysis in Several Variables, x+208 pp. (1966) · Zbl 0138.06203
[14] Kolodziej, Slawomir, The complex {M}onge-{A}mp\`{e}re equation, Acta Math.. Acta Mathematica, 180, 69-117 (1998) · Zbl 0913.35043 · doi:10.1007/BF02392879
[15] Phong, Duong H.; Sesum, Natasa; Sturm, Jacob, Multiplier ideal sheaves and the {K}\"{a}hler-{R}icci flow, Comm. Anal. Geom.. Communications in Analysis and Geometry, 15, 613-632 (2007) · Zbl 1143.53064 · doi:10.4310/CAG.2007.v15.n3.a7
[16] Phong, Duong H.; Picard, Sebastien; Zhang, Xiangwen, Fu-{Y}au {H}essian equations, J. Differential Geom.. Journal of Differential Geometry, 118, 147-187 (2021) · Zbl 1473.58015 · doi:10.4310/jdg/1620272943
[17] Phong, Duong H.; T\^o, Dat T., Fully non-linear parabolic equations on compact {H}ermitian manifolds, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`{e}me S\'{e}rie, 54, 793-829 (2021) · Zbl 07452925 · doi:10.24033/asens.2471
[18] Song, Jian; Tian, Gang, The {K}\"{a}hler-{R}icci flow through singularities, Invent. Math.. Inventiones Mathematicae, 207, 519-595 (2017) · Zbl 1440.53116 · doi:10.1007/s00222-016-0674-4
[19] Sz\'{e}kelyhidi, G\'{a}bor, Fully non-linear elliptic equations on compact {H}ermitian manifolds, J. Differential Geom.. Journal of Differential Geometry, 109, 337-378 (2018) · Zbl 1409.53062 · doi:10.4310/jdg/1527040875
[20] Tian, Gang, On {K}\"{a}hler-{E}instein metrics on certain {K}\"{a}hler manifolds with {\(C_1(M)>0\)}, Invent. Math.. Inventiones Mathematicae, 89, 225-246 (1987) · Zbl 0599.53046 · doi:10.1007/BF01389077
[21] Tosatti, Valentino, Adiabatic limits of {R}icci-flat {K}\"{a}hler metrics, J. Differential Geom.. Journal of Differential Geometry, 84, 427-453 (2010) · Zbl 1208.32024 · doi:10.4310/jdg/1274707320
[22] Tosatti, Valentino; Weinkove, Ben, The complex {M}onge-{A}mp\`{e}re equation with a gradient term, Pure Appl. Math. Q.. Pure and Applied Mathematics Quarterly, 17, 1005-1024 (2021) · Zbl 1477.32061 · doi:10.4310/PAMQ.2021.v17.n3.a7
[23] Wang, Jiaxiang; Wang, Xu-Jia; Zhou, Bin, A priori estimate for the complex {M}onge-{A}mp\`{e}re equation, Peking Math. J.. Peking Mathematical Journal, 4, 143-157 (2021) · Zbl 1477.32062 · doi:10.1007/s42543-020-00025-3
[24] Yau, Shing Tung, On the {R}icci curvature of a compact {K}\"{a}hler manifold and the complex {M}onge-{A}mp\`{e}re equation. {I}, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.