An analytic proof of Griffiths’ conjecture on compact Riemann surfaces

Griffiths’ conjecture asserts that a holomorphic vector bundle is ample if and only if it admits a Hermitian metric with positive curvature. In this paper, we present a new proof of this conjecture on compact Riemann surfaces using a system of PDEs introduced by Demailly. Our argument combines techniques developed by Uhlenbeck–Yau for Hermitian–Einstein metrics with Pingali’s reduction of the problem to an a priori estimate.