Function Theory of pseudoconvex subdomains and Geometry of boundaries in Kahler manifolds

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The purpose of this research is to show the relation with pseudoconvex subdomains and its boundary in Kahler manifolds by using differential-geometric methods. The main tool is the Levi form to the boundary and we studied it explicitly.The results are the following.(1) In the case M = C^n , we write explicitly the Levi form of the distance function to a complex submanifold S in C^n by using the defining function of S. As its application, we found the relation with the conditoin for the Levi form to degenerate in complex tangential condition and the condition for the submanifold S to be developable.(2) In the case M = C^2, we write explicitly the Levi form of the distance function to a real submanifold S in C^2 by using the defining function of S. As its application, we discussed the example of pseudoconvex domains in complex tori showed by Grauert.(3) In the case M = C^n , we write explicitly the Levi form of the distance function to a real submanifold S in C^n by using the defining function of S, and give a necessary and sufficient condition for the Levi form to degenrrate.