Concavity property of minimal $L^{2}$ integrals with Lebesgue measurable gain VIII -- partial linearity and log-convexity

TL;DR

This paper investigates the conditions under which the concavity of minimal L^2 integrals degenerates to partial linearity, exploring its relation to log-convexity of Bergman kernels on Riemann surfaces.

Contribution

It provides necessary conditions and characterizations for the degeneracy of concavity to partial linearity, linking it to log-convexity properties of Bergman kernels.

Findings

Necessary conditions for concavity degenerating to partial linearity.

Characterization of this degeneracy on open Riemann surfaces.

Relations between minimal L^2 integrals' concavity and Bergman kernel log-convexity.

Abstract

In this article, we give some necessary conditions for the concavity property of minimal $L^2$ integrals degenerating to partial linearity, a charaterization for the concavity degenerating to partial linearity for open Riemann surfaces, and some relations between the concavity property for minimal $L^2$ integrals and the log-convexity for Bergman kernels.