Curvatures of metrics in non-compact complex manifolds

Abstract : We study the relationships between geometric properties and metric properties of domains in C^n.More precisely, we are interested in the behavior of holomorphic bisectional curvatures of invariant Kähler metrics, namely the Bergman metric and the Kähler-Einstein metric, near the boundary of bounded pseudoconvex domains with smooth boundary.We prove that at boundary points that are either strictly pseudoconvex or such that the squeezing function of the domain tends to one the holomorphic bisectional curvatures of the Kähler-Einstein metric of the domain tends to the holomorphic bisectional curvatures of the Kähler-Einstein metric of the ball.We also study the holomorphic bisectional curvatures of the Kähler-Einstein metric and of the Bergman metric in some polynomial domains (namely tube and Thullen domains in C^2) which serve as local models at boundary point of finite type. Using these studies we prove that at certain boundary points of smoothly bounded convex domains of finite type there exists a non tangential neighbourhood such the holomorphic bisectional curvatures of the Kähler-Einstein metric are pinched between two negative constants. We also prove that for every smoothly bounded pseudoconvex complete Reinhardt domain of finite type inf C^2 there exists a neighbourhood of the boundary relative to the domain in which the holomorphic bisectional curvatures of the Bergman metric are pinched between two negative constants.
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Sébastien Gontard. Curvatures of metrics in non-compact complex manifolds. Differential Geometry [math.DG]. Université Grenoble Alpes, 2019. English. ⟨NNT : 2019GREAM027⟩. ⟨tel-02307850⟩

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