Infinite geodesic rays in the space of Kähler potentials
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 617-630.

In this paper we prove the existence of solutions of a degenerate complex Monge-Ampére equation on a complex manifold. Applying our existence result to a special degeneration of complex structure, we show how to associate to a change of complex structure an infinite length geodetic ray in the space of potentials. We also prove an existence result for the initial value problem for geodesics. We end this paper with a discussion of a list of open problems indicating how to relate our reults to the existence problem for extremal metrics.

Classification : 32Q15, 32W20, 58D27, 58E11 
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     title = {Infinite geodesic rays in the space of {K\"ahler} potentials},
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Arezzo, Claudio; Tian, Gang. Infinite geodesic rays in the space of Kähler potentials. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 617-630. http://www.numdam.org/item/ASNSP_2003_5_2_4_617_0/

[1] D. Burns, Curvatures of Monge-Ampère foliations and parabolic manifolds, Ann. of Math. (2) 115 (1982), 349-373. | MR | Zbl

[2] E. Calabi, Extremal Kähler metrics, In: “Seminar on Differential Geometry”, Ann. Math. Stud. 102 (1982), 259-290. | MR | Zbl

[3] E. Calabi, Extremal Kähler metrics, II, In: “Differential Geometry and Complex analysis”, Lecture Notes in Math., Springer, 1985, 96-114. | MR | Zbl

[4] E. Calabi - X. Chen , The space of Kähler metrics II, J. Differential Geom. 61 (2002), 173-193. | MR | Zbl

[5] X. Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000), 189-234. | MR | Zbl

[6] X. Chen, On the lower bound of the Mabuchi energy and its application, Internat. Math. Res. Notices 12 (2000), 607-623. | MR | Zbl

[7] X. Chen, Recent progress in Kähler Geometry, In: “Proc. of the Internat Congress of Mathematicians 2002”, Vol. II, 273-282. | MR | Zbl

[8] S. Y. Cheng - S. T. Yau, Inequality between Chern numbers of singular Kähler surfaces and characterization of orbit space of discrete group of SU(2,1), Contemp. Math. 49 (1986), 31-43. | MR | Zbl

[9] W. Ding - G. Tian G, Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math. 110 (1992), 315-335. | MR | Zbl

[10] S. K. Donaldson, Remarks on gauge theory, complex geometry, and 4-manifold topology, In: “The Fields Medal volume”, World Scientific, 1997. | MR

[11] S. K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, In: “Northern California Symplectic Geometry Seminar”, 13-33, Amer. Math. Soc. Transl. (2) 196 Amer. Math. Soc., Providence, RI, 1999. | MR | Zbl

[12] S. K. Donaldson, Scalar curvature and projective embeddings I, J. Differential Geom. 59 (2001), 479-522. | MR | Zbl

[13] S. K. Donaldson, Holomorphic disks and the complex Monge-Ampere equation, to appear in J. Symplectic Geometry. | MR | Zbl

[14] D. Guan, On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles, Math. Res. Lett. 6 (1999), 547-555. | MR | Zbl

[15] V. Guillemin - M. Stenzel, Grauert tubes and the homogeneous Monge-Ampère equation, J. Differential Geom. 34 (1991), 561-570. | MR | Zbl

[16] C. Lebrun, Polarized 4-manifolds, extremal Kähler metrics, and Seiberg-Witten theory, Math. Res. Lett. 5 (1995), 653-662. | MR | Zbl

[17] Z. Lu, On the lower terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), 235-273. | MR | Zbl

[18] T. Mabuchi, K-energy maps integrating Futaki invariants, Tohoku Math. J. (1986), 575-593. | MR | Zbl

[19] T. Mabuchi T, Some symplectic geometry on compact Kähler manifolds, Osaka J. Math. 24 (1987), 227-252. | MR | Zbl

[20] S. Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), 495-550. | MR | Zbl

[21] G. Tian, On Kähler - Einstein metrics on certain Kähler manifolds with C(M)>0, Invent. Math. 89 (1987), 225-246. | MR | Zbl

[22] G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99-130. | MR | Zbl

[23] G. Tian, On Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1-37. | MR | Zbl

[24] G. Tian - S. T. Yau, Complete Kähler manifolds with zero Ricci curvature, Invent. Math. 106 (1991), 27-60. | MR | Zbl

[25] S. Zeldtich, Szegö Kernel and a Theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317-331. | MR | Zbl