Complete Ricci-flat metrics through a rescaled exhaustion

الأصل · EN

Typical existence result on Ricci-flat metrics is in manifolds of finite geometry, that is, on F= F-D where F is a compact Kähler manifold and D is a smooth divisor. We view this existence problem from a different perspective. For a given complex manifold X, we take a suitable exhaustion {Xᵣ}ᵣ>₀ admitting complete s of negative Ricci. Taking a positive decreasing sequence {λᵣ}ᵣ>₀, ᵣ→∞λᵣ=0, we rescale the metric so that gᵣ is the complete in Xᵣ of Ricci curvature -λᵣ. The idea is to show the limiting metric ᵣ→∞ gᵣ does exist. If so, it is a Ricci-flat metric in X. Several examples: X=Cⁿ and X=TM where M is a compact rank-one symmetric space have been studied in this article. The existence of complete s of negative Ricci in bounded domains of holomorphy is well-known. Nevertheless, there is very few known for unbounded cases. In the last section we show the existence, through exhaustion, of such kind of metric in the unbounded domain TπHⁿ.