Quillen connection and uniformization of Riemann surfaces

The Quillen connection on L -> M-g, where L* the Hodge line bundle over the moduli stack of smooth complex projective curves curves M-g, g >= 5, is uniquely determined by the condition that its curvature is the Weil-Petersson form on M-g. The bundle of holomorphic connections on E has a unique holomorphic isomorphism with the bundle on M-g given by the moduli stack of projective structures. This isomorphism takes the C-infinity section of the first bundle given by the Quillen connection on E to the C-infinity section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.