Variation of Bergman metrics on Riemann surfaces

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Hodge metrics and the curvature of higher direct images (2007, unknown): “Ko2] for further remarks on related results. On the analytic side, it can be understood as the plurisubharmonic variation of related functions to the Robin constant [Y][LY], or of the Bergman kernels [MY]. There is also a series of works by Yamaguchi. As he mentioned in [B], his method is inspired b”
A proof of the Ohsawa–Takegoshi theorem with sharp estimates (2014, background): “Since D := {(τ, z);G(z)− Re τ < 0} is pseudoconvex inC(2), it follows from [13], [1] thatlogKt(0) is a convex function of t.”
Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics (2014, result): “ressed that one of the main points of our approach is that it does not require the existence of any sort of Monge-Ampère foliation. Our proof uses plurisubharmonic variation of local Bergman kernels ([41], [11]); see Section 1.2 below for a sketch of the proof and Section 3.2 for comparison with pre”
Bergman kernels and the pseudoeffectivity of relative canonical bundles (2007, background): “Several versions of the theorem above were established by the first named author in his series of articles on the plurisubharmonic variation of the Bergman kernels (see [1], [2], [3] and also [22] for the first results in this direction).”