Hölder continuity of solutions to the complex Monge-Ampère equation with the right-hand side in \(L^{p}\): the case of compact Kähler manifolds. (English) Zbl 1149.32018

Let \(M\) be a compact \(n\)-dimensional Kähler manifold with the fundamental form \(\omega\). On this manifold the author considers a solution of the Monge-Ampère equation
\[ (\omega + d d^c u)^n = f \omega^n, \]
where \(f \in L^p (M), p > 1; f \geq 0;\)
\[ \int_M f \omega^n = \int_M \omega^n \]
is a given function. He proves that the solution \(u\) of this equation is Hölder continuous (Theorem 2.1).


32U05 Plurisubharmonic functions and generalizations
32U40 Currents
32W20 Complex Monge-Ampère operators
Full Text: DOI arXiv


[1] Bedford E., Taylor B.A.: The Dirichlet problem for the complex Monge–Ampère operator. Invent. math. 37, 1–44 (1976) · Zbl 0325.31013
[2] Demailly J.-P.: Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1, 361–409 (1992) · Zbl 0777.32016
[3] Guedj, V., Kołodziej, S., Zeriahi, A.: Hölder continuous solutions to the complex Monge–Ampère equations. math.CV/0607314 (2006) · Zbl 1296.32012
[4] Eysssidieuxm, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. math.AG/0603431 (2006)
[5] Kołodziej S.: The complex Monge–Ampère equation. Acta Math. 180, 69–117 (1998) · Zbl 0913.35043
[6] Kołodziej S.: The Monge–Ampere equation on compact Kähler manifolds. Indiana U. Math. J. 52, 667–686 (2003) · Zbl 1039.32050
[7] Kołodziej S.: The complex Monge–Ampère equation and pluripotential theory. Mem. AMS 840, 62 (2005) · Zbl 1084.32027
[8] Song J., Tian G.: The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170, 609–653 (2007) · Zbl 1134.53040
[9] Tian G., Zhang Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. B 27(2), 179–192 (2006) · Zbl 1102.53047
[10] Yau S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. Commun. Pure Appl. Math. 31, 339–411 (1978) · Zbl 0369.53059
[11] Zhang, Z.: On Degenerated Monge–Ampere Equations over Closed Kähler Manifolds. IMRN, vol. 2006, pp. 1–18 (2006)
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