Kähler manifolds with curvature bounded from above by a decreasing function
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- by Mitsuhiro Itoh
- Proc. Amer. Math. Soc. 75 (1979), 289-293
- DOI: https://doi.org/10.1090/S0002-9939-1979-0532153-8
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Abstract:
Let M be a simply connected complete Kähler manifold. If M has curvature bounded from above by a certain positive decreasing function, then it is a Stein manifold, diffeomorphic to a euclidean space. This fact is a generalization of the well-known propositions for complete manifolds of nonpositive curvature and is shown by the aid of a Rauch comparison theorem for conjugate points together with a comparison theorem of Siu and Yau with respect to the Hessian of distance functions.References
- É. Cartan, Leçon sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris, 1963.
- Paul F. Klembeck, A complete Kähler metric of positive curvature on $C^{n}$, Proc. Amer. Math. Soc. 64 (1977), no. 2, 313–316. MR 442290, DOI 10.1090/S0002-9939-1977-0442290-2
- Shoshichi Kobayashi, Riemannian manifolds without conjugate points, Ann. Mat. Pura Appl. (4) 53 (1961), 149–155. MR 150700, DOI 10.1007/BF02417792
- Yum Tong Siu and Shing Tung Yau, Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay, Ann. of Math. (2) 105 (1977), no. 2, 225–264. MR 437797, DOI 10.2307/1970998 H. H. Wu, Negatively curved Kähler manifolds, Notices Amer. Math. Soc. 14 (1967), 515.
Bibliographic Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 75 (1979), 289-293
- MSC: Primary 53C20; Secondary 32E10, 32F30, 53C55
- DOI: https://doi.org/10.1090/S0002-9939-1979-0532153-8
- MathSciNet review: 532153