A Note on variation of bergman metrics on riemann surfaces under pseudoconvexity

Abstract

We study the variation of the Bergman disk δϵ(t) with center ζ(t) and radius ϵ>0 on the moving Riemann surface R(t) with parameter t in a disk B, and show that, if the total space is a strictly pseudoconvex domain, then, for sufficiently small ϵ > 0, the domain is also pseudoconvex in case ζ(t) is holomorphic on B.