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Hölder continuous potentials on manifolds with partially positive curvature

Part of: Pluripotential theory

Published online by Cambridge University Press:  16 April 2010

Sławomir Dinew
Sławomir Dinew
Affiliation:
Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland, (slawomir.dinew@im.uj.edu.pl)
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Abstract

It is proved that solutions of the complex Monge–Ampère equation on compact Kähler manifolds with right hand side in Lp, p > 1, are uniformly Hölder continuous under the assumption on non-negative orthogonal bisectional curvature.

Keywords

Kähler manifoldbisectional curvaturecomplex Monge–Ampère equation

MSC classification

Secondary: 32U15: General pluripotential theory

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 4 , October 2010 , pp. 705 - 718
Copyright
Copyright © Cambridge University Press 2010

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References

1.Błocki, Z., Regularity of the degenerate Monge–Ampére equation on compact Kähler manifolds, Math. Z. 244 (2003), 153–161.CrossRefGoogle Scholar
2.Błocki, Z., A gradient estimate in Calabi–Yau theorem, Math. Annalen 344 (2009), 317–327.CrossRefGoogle Scholar
3.Błocki, Z. and Kołodziej, S., Regularization of plurisubharmonic functions on manifolds, Proc. Am. Math. Soc. 135(7) (2007), 2089–2093.CrossRefGoogle Scholar
4.Cascini, P. and Nave, G. La, Kähler–Ricci flow and the minimal model program, preprint (arXiv math.AG/0603064).Google Scholar
5.Chen, X. X., On Kähler manifolds with positive orthogonal bisectional curvature, Adv. Math. 215(2) (2007), 427–445.CrossRefGoogle Scholar
6.Demailly, J.-P., A numerical criterion for very ample line bundles, J. Diff. Geom. 37 (1993), 323–374.Google Scholar
7.Demailly, J.-P., Regularization of closed positive currents of type (1, 1) by the flow of a Chern connection, Aspects of Mathematics, Volume E26, pp. 105–126 (Vieweg, 1994).Google Scholar
8.Demailly, J.-P., Complex analytic and differential geometry (self-published e-book accessible at www-fourier.ujf-grenoble.fr/.demailly).Google Scholar
9.Dinew, S. and Zhang, Z., On stability and continuity of bounded solutions of degenerate complex Monge–Ampère equations over compact Kähler manifolds, Adv. Math. in press (doi: 10.1016/j.aim.2010.03.001).Google Scholar
10.Eyssidieux, P., Guedj, V. and Zeriahi, A., Singular Kähler–Einstein metrics, J. Am. Math. Soc. 22 (2009), 607–639.CrossRefGoogle Scholar
11.Gu, H.-L. and Zhang, Z.-H., An extension of Mok's theorem on the generalized Frankel conjecture, preprint (arXiv:0709.4086v3).Google Scholar
12.Guedj, V., Kołodziej, S. and Zeriahi, A., Hölder continuous solutions to Monge–Ampère equations, Bull. Lond. Math. Soc. 40 (2008), 1070–1080.CrossRefGoogle Scholar
13.Kołodziej, S., The complex Monge–Ampère equation, Acta Math. 180 (1998), 69–117.CrossRefGoogle Scholar
14.Kołodziej, S., The Monge–Ampére equation on compact Kähler manifolds, Indiana Univ. Math. J. 52(3) (2003), 667–686.CrossRefGoogle Scholar
15.Kołodziej, S., The complex Monge–Ampère equation and pluripotential theory, Memoirs of the American Mathematical Society, Volume 178/840, pp. 1–64 (American Mathematical Society, Providence, RI, 2005).Google Scholar
16.Kołodziej, S., Hölder continuity of solutions to the complex Monge–Ampère equation with the right hand side in Lp, Math. Annalen 342 (2008), 379–386.CrossRefGoogle Scholar
17.Pliś, Sz., A counterexample to the regularity of the degenerate complex Monge–Ampère equation, Ann. Polon. Math. 86 (2005), 171–175.CrossRefGoogle Scholar
18.Siu, Y.-T., Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics, DMV seminar (Birkhäuser, 1987).Google Scholar
19.Song, J. and Tian, G., The Kähler–Ricci flow on surfaces of positive Kodaira dimension, Invent. Math. 170 (2007), 609–653.CrossRefGoogle Scholar
20.Song, J. and Tian, G., Canonical measures and the Kähler–Ricci flow, preprint (arXiv 0802.2570).Google Scholar
21.Tian, G. and Zhang, Z., On the Kähler–Ricci flow on projective manifolds of general type, Chin. Annals Math. 27(2) (2006), 179–192.CrossRefGoogle Scholar