CURVATURE ESTIMATE AND THE RAMIFICATION OF THE HOLOMORPHIC MAPS OVER HYPERSURFACES ON RIEMANN SURFACES

In this paper, we give a Gauss curvature estimate on an open Riemann surface S whose metric is of the form ds(2) = parallel to G parallel to(2m) vertical bar omega vertical bar(2), where omega is a holomorphic 1-form, m is an element of Z(+) and G is a reduced representation of a holomorphic map g from S into a projective subvariety V of P-n(C) that is ramified over a family of hypersurfaces in N-subgeneral position with respect to V.