An approach to Griffiths conjecture

The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on ${\mathcal{O}}_{\mathbb{P}(E^{\ast })}(1)$ to induce a Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests to study the relative K"ahler-Ricci flow on ${\mathcal{O}}_{\mathbb{P}(E^{\ast })}(1)$ for the fibration $\mathbb{P}(E^{\ast })\to S$. We define a flow and give arguments for the convergence.