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A curvature formula associated to a family of pseudoconvex domains (1508.00242v4)
Published 2 Aug 2015 in math.CV and math.DG
Abstract: We shall give a definition of the curvature operator for a family of weighted Bergman spaces ${\mathcal H_t}$ associated to a smooth family of smoothly bounded strongly pseudoconvex domains ${D_t}$. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries ${\partial D_t}$. As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for ${\mathcal H_t}$ and its applications to triviality of fibrations are also given in this paper.