A priori estimates for weak solutions of complex Monge-Ampère equations

Benelkourchi, Slimane; Guedj, Vincent; Zeriahi, Ahmed

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 1, pp. 81-96.

Let $X$ be a compact Kähler manifold and $\omega$ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${ℰ}_{\chi }(X,\omega )$ of $\omega$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${ℰ}_{\chi }(X,\omega )$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends those of U. Cegrell’s and S. Kolodziej’s and puts them into a unifying frame. It also gives a simple proof of S. T. Yau’s celebrated a priori ${𝒞}^0$-estimate.

[1] H. Alexander and B. A. Taylor, Comparison of two capacities in ${ℂ}^n$, Math. Z. 186 (1984), 407-417. | EuDML | MR | Zbl

[2] T. Aubin, Équations du type Monge-Ampère sur les variétés Kählériennes compactes, Bull. Sci. Math. (2) 102 (1978), 63-95. | MR | Zbl

[3] E. Bedford and B. A. Taylor A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. | MR | Zbl

[4] Z. Blocki, On uniform estimate in Calabi-Yau theorem, Sci. China Ser. A 48 (2005) , suppl., 244-247. | MR | Zbl

[5] G. Burgos, J. Kramer and U. Kuhn, Arithmetic characteristic classes of automorphic vector bundles, Doc. Math. 10 (2005), 619-716. | EuDML | MR | Zbl

[6] E. Calabi, On Kähler manifolds with vanishing canonical class. Algebraic geometry and topology, In: “A symposium in Honor of S. Lefschetz”, Princeton Univ. Press, Princeton, N. J. (1957), 78-89. | MR | Zbl

[7] U. Cegrell, Pluricomplex energy, Acta Math. 180 (1998), 187-217. | MR | Zbl

[8] P. Eyssidieux, V. Guedj and A. Zeriahi, Singular Kähler-Einstein metrics, preprint arxiv math. AG/0603431. | MR | Zbl

[9] V. Guedj and A. Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), 607-639. | MR | Zbl

[10] V. Guedj and A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), 442-482. | MR | Zbl

[11] S. Kolodziej, The range of the complex Monge-Ampère operator, Indiana Univ. Math. J. 43 (1994), 1321-1338. | MR | Zbl

[12] S. Kolodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998), 69-117. | MR | Zbl

[13] S. Kolodziej, The Monge-Ampère equation on compact Kähler manifolds, Indiana Univ. Math. J. 52 (2003), 667-686. | MR | Zbl

[14] S. Kolodziej, “The Complex Monge-Ampère Equation and Pluripotential Theory”, Mem. Amer. Math. Soc., Vol. 178, 2005. | MR | Zbl

[15] U. Kuhn, Generalized arithmetic intersection numbers, J. Reine Angew. Math. 534 (2001), 209-236. | MR | Zbl

[16] J. Rainwater, A note on the preceding paper, Duke Math. J. 36 (1969), 799-800. | MR | Zbl

[17] T. Ransford, “Potential Theory in the Complex Plane”, London Mathematical Society Student Texts, Vol. 28, Cambridge University Press, Cambridge, 1995. | MR | Zbl

[18] G. Tian, “Canonical Metrics in Kähler Geometry”, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2000. | MR | Zbl

[19] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31 (1978), 339-411. | MR | Zbl

[20] A. Zeriahi, Volume and capacity of sublevel sets of a Lelong class of psh functions, Indiana Univ. Math. J. 50 (2001), 671-703. | MR | Zbl