Approximation of weak geodesics and subharmonicity of Mabuchi energy

Chen, XiuXiong; Li, Long; Păuni, Mihai

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 5, pp. 935-957.

Dans cet article nous obtenons des résultats concernant l’approximation des géodésiques qui connectent deux métriques kähleriennes dans la même classe de cohomologie. Comme corollaire, nous obtenons une preuve de la convexité de la fonctionnelle de Mabuchi le long des géodésiques. C’est un théorème obtenu récemment par Berman-Berndtsson, et nos arguments représentent une version « globale » de leur démonstration originale.

In this paper we are interested in the approximation of weak geodesics connecting two Kähler metrics in the same cohomology class. As a consequence, we derive the convexity of the Mabuchi energy along geodesics. This important result was obtained recently by Berman-Berndtsson, and our approach can be seen as a “global version” of their original proof.

[1] Berman (R.).— A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics, arXiv:1011.3976. | DOI | MR | Zbl

[2] Berman (R.), Boucksom (S.), Guedj (V.), Zeriahi (A.).— A variational approach to complex Monge-Ampère equations, Publications mathématiques de l’IHES, 117(1), p. 179-245 (2013). | DOI

[3] Berman (R.), Demailly (J.-P.).— Regularity of plurisubharmonic upper envelopes in big cohomology classes, Proceedings of the Symposium “Perspectives in Analysis, Geometry and Topology” in honor of Oleg Viro (Stockholm University, May 2008), Progress in Math. 296, Birkhäuser/Springer, New York (2012). | DOI | Zbl

[4] Berman (R.), Berndtsson (B.).— Convexity of the K-energy on the space of Kähler metrics, arXiv:1405.0401 | DOI

[5] Blocki (Z.).— A gradient estimate in the Calabi-Yau theorem, Math. Ann., 344(2), p. 317-327 (2009). | DOI | MR | Zbl

[6] Blocki (Z.).— On geodesics in the space of Kähler metrics, Proceedings of the “Conference in Geometry" dedicated to Shing-Tung Yau (Warsaw, April 2009), in “Advances in Geometric Analysis", ed. S. Janeczko, J. Li, D. Phong, Advanced Lectures in Mathematics 21, International Press, p. 3-20 (2012). | Zbl

[7] Chen (X.-X.).— Space of Kähler metrics. Journal of Differential Geometry, 56(2), p. 189-234 (2000). | DOI | Zbl

[8] Chen (X.-X.).— On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 2000, no. 12, p. 607-623. | DOI | MR | Zbl

[9] Chen (X.-X.), Tian (G.).— Geometry of Kähler metrics and foliations by holomorphic discs, Publications Mathématiques de l’IHES, Vol 107, p. 1-107 (2008). | DOI | Zbl

[10] Darvas (T.), Lempert (L.).— Weak geodesics in the space of Kähler metrics, Math. Res. Lett. 19, no. 5, p. 1127-1135 (2012). | DOI | Zbl

[11] Demailly (J.-P.).— Regularization of closed positive currents of type (1,1) by the flow of a Chern connection, Actes du Colloque en l’honneur de P. Dolbeault (Juin 1992), édité par H. Skoda et J.M. Trépreau, Aspects of Mathematics, Vol. E 26, Vieweg, p. 105-126 (1994). | DOI | Zbl

[12] Gilbarg (D.), Trudinger (N.).— Elliptic Partial Differential Equation of Second Order, Springer. | DOI

[13] Greene (R. E.), Wu (H.).— $C^{\infty }$ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. Ecole Norm. Sup. (4) 12, no. 1 p. 47-84 (1979). | DOI | Zbl

[14] Guedj (V.), Zeriahi (A.).— The weighted Monge-Ampère energy of quasipsh functions, J. Funct. An. 250, p. 442-482 (2007). | DOI | Zbl

[15] Kolodziej (S.).— Hölder continuity of solutions to the complex Monge-Ampère equation with the right-hand side in $L^p$: the case of compact Kähler manifolds, Math. Ann. 342, no. 2, p. 379-386 (2008). | DOI | Zbl

[16] Lempert (L.), Vivas (L.).— Geodesics in the space of Kähler metrics, Duke Math. J. 162, no. 7, p. 1369-1381 (2013). | DOI | Zbl

[17] Păun (M.).— Relative adjoint transcendental classes and Albanese maps of compact Kaehler manifolds with nef Ricci curvature, arXiv:1209.2195 | DOI | Zbl