Variation of Bergman metrics on Riemann surfaces

We study the variation of the Berman metric K(t,)|d|² on the Riemann surface R(t) (which varies with complex parameter t in a disk B) such that :B is a 2-dimensional Stein space with ^–1(t)=R(t),t B. We show that log K(t,)|d|² is plurisubharmonic in , and apply it to give the uniformization condition for the special Stein space.Revised version: 18 February 2004Mathematics Subject Classification (2000): 30C85, 30F15, 32F05, 32H10