### Minimal surfaces and deformations of holomorphic curves in Kähler-Einstein manifolds

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In this paper we prove the existence of solutions of a degenerate complex Monge-Ampére equation on a complex manifold. Applying our existence result to a special degeneration of complex structure, we show how to associate to a change of complex structure an infinite length geodetic ray in the space of potentials. We also prove an existence result for the initial value problem for geodesics. We end this paper with a discussion of a list of open problems indicating how to relate our reults to the...

We consider Fano manifolds $M$ that admit a collection of finite automorphism groups ${G}_{1},...,{G}_{k}$ , such that the quotients $M/{G}_{i}$ are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that $M$ admits a Kähler-Einstein metric too.

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