Abstract. Let X be a compact Kähler manifold and S a subvariety of X with higher co-dimension. The aim is to study complete constant scalar curvature Kähler metrics on non-compact Kähler manifold X − S with Poincaré-MokYau asymptotic property (see Definition 1.1). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.

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Let M be a compact Kähler manifold and N be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler-Einstein metrics on M − N . As an application, let E be an exceptional divisor of M . Then M − E cannot admit any complete Kähler-Einstein metric if blow-down of E is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs. Problem 1.1. Let M be a compact Kähler manifold and N be a subvariety with codimension bigger than or equal to 2, how to find a complete canonical metric on the noncompact Kähler manifold M − N ? Date: March 22, 2016.

The main purpose of this paper is to study the properties of transversally harmonic maps by using Bochner-type formulas. As an application, we obtain the following theorem between compact Sasaki manifolds: Let f be a transversally harmonic map from compact Sasaki manifold M to compact Sasaki manifold M , and M has a strongly negative transverse curvature. If the rank of d T f is at least three at some points of M , then f is contact holomorphic (or contact anti-holomorphic).

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