Abstract
In this paper we shall give a survey of the known results and methods concerning the strong rigidity of Kähler manifolds and present some new related results. The important phenomenon of strong rigidity was discovered by Professor G.D. Mostow in the case of locally symmetric nonpositively curved Riemannian manifolds. He proved [18] that two compact locally symmetric nonpositively curved Riemannian manifolds are isometric up to normalization constants if they have the same fundamental group and neither one contains a closed one or two dimensional totally geodesic submanifold that is locally a direct factor. This last assumption is clearly necessary because of the existence of non-trivial holomorphic deformations of any compact Riemann surface of genus at least two. Mostow’s result says that if one can rule out the possibility of contribution to the change of metric structure from certain submanifolds of dimension two or lower, the metric structure is rigidly determined by the topology for compact locally symmetric nonpositively curved manifolds. Mostow’s result also holds for the non-compact complete case under the assumption of finite volume.
Research partially supported by a National Science Foundation grant.
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Siu, YT. (1987). Strong Rigidity for Kähler Manifolds and the Construction of Bounded Holomorphic Functions. In: Howe, R. (eds) Discrete Groups in Geometry and Analysis. Progress in Mathematics, vol 67. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6664-3_5
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