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Complete Kähler manifolds with zero Ricci curvature. II. (English) Zbl 0766.53053
This is a continuation of the previous paper [J. Am. Math. Soc. 3, 579–609 (1990; Zbl 0719.53041)] on settling the non-compact version of Calabi’s conjecture on quasiprojective manifolds \(M\) which can be written as \(\overline{M}\setminus D\). The whole arguments in the previous paper are generalized to the case that \(\overline{M}\) is a normal Kähler orbifold and \(D\) is an admissible divisor. The authors construct complete Kähler metrics on \(M=\overline{M}\setminus D\) with prescribed Ricci form in \(C_ 1(-K_{\overline M}-\beta L_ D)\) for \(\beta>1\) under some suitable assumptions on \(\overline{M}\) and \(D\). They find complete Kähler metrics on \(M\) with either zero Ricci curvature of finite topological type or non-negative Ricci curvature. These constructions include practically all known examples of complete Kähler manifolds with zero Ricci curvature of finite topological type. Besides constructing many new examples of such manifolds which may serve as gravitational instantons, these metrics provide a bridge between metric geometry and algebraic geometry of \(M\) because the authors do have some understanding of complete manifolds with non-negative curvature.

53C55 Global differential geometry of Hermitian and Kählerian manifolds
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[1] [Ba] Baily, W.: On the imbedding ofV-manifolds in projective space. Am. J. Math.79, 403-430 (1957) · Zbl 0173.22706 · doi:10.2307/2372689
[2] [BK] Bando, S., Kobayashi, R.: Complete Ricci-flat Kähler metrics. (Preprint)
[3] [CY1] Cheng, S.Y., Yau, S.T.: On the existence of complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math.33, 507-544 (1980) · Zbl 0506.53031 · doi:10.1002/cpa.3160330404
[4] [CY2] Cheng, S.Y., Yau, S.T.: Inequality between Chern numbers of singular Kähler surfaces and characterization of orbit space of discrete group of SU (2, 1). Contemp. Math.49, 31-43 (1986)
[5] [Fed] Federer, H.: Geometry Measure Theory. (Grundlehren Math. Wiss., Bd. 153) Berlin Heidelberg New York: Springer 1969
[6] [Fef] Fefferman, C.: Monge-Ampère equations, the Berman kernel, and geometry of pseudoconvex domains. Ann. Math.103, 395-416 (1976) · Zbl 0322.32012 · doi:10.2307/1970945
[7] [Gr] Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math., Inst. Hautes Étud. Sci.53 (1981) · Zbl 0474.20018
[8] [GT] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin Heidelberg New York: Springer 1977 · Zbl 0361.35003
[9] [Ko] Kobayashi, S.: On compact Kähler manifolds with positive definite Ricci tensor. Ann. Math.74, 570-574 (1961) · Zbl 0107.16002 · doi:10.2307/1970298
[10] [Sh] Shiffman, B.: Vanishing theorems on complex manifolds. (Prog. Math., vol. 56) Basel Boston Stuttgart: Birkhaüser 1985 · Zbl 0578.32055
[11] [Si] Simon, L.: Lectures on geometric measure theory. Proc. Cent. Math. Anal. Aust. Natl. Univ.3 (1983) · Zbl 0546.49019
[12] [T1] Tian, G.: On Kähler-Einstein metrics on certain Kähler manifolds with C1(M)>0. Invent. Math.89, 225-246 (1987) · Zbl 0599.53046 · doi:10.1007/BF01389077
[13] [T2] Tian, G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math101, 101-172 (1990) · Zbl 0716.32019 · doi:10.1007/BF01231499
[14] [TY1] Tian, G., Yau, S.T.: Complete Kähler manifolds with zero Ricci curvature. 1. J. Am. Math. Soc.3, 579-610 (1990) · Zbl 0719.53041
[15] [TY2] Tian, G., Yau, S.T.: Kähler-Einstein metrics on complex surfaces with C1>0. Commun. Math. Phys.112, 175-203 (1987) · Zbl 0631.53052 · doi:10.1007/BF01217685
[16] [TY3] Tian, G., Yau, S.T.: (Preprint)
[17] [Y1] Yau, S.T.: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. Éc. Norm. Supér., IV. Sér.8, 487-507 (1975) · Zbl 0325.53039
[18] [Y2] Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I*. Commun. Pure Appl. Math.31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
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