ON THE LOWER BOUND ESTIMATES OF SECTIONS OF THE CANONICAL BUNDLES OVER A RIEMANN SURFACE

We give a lower bound estimate of the sum of the square norm of the sections of the pluricanonical bundles over a Riemann surface of genus greater than 2 and Gauss curvature -1. Such an estimate must depend on the injective radius of the Riemann surface. However, using this estimate, we give a uniform estimate of the corona problem on Riemann surface. Here "uniform" means that the estimate depends only on the genus of Rieman surface, not on the injective radius.