On vector bundles of finite order

We define Finsler metrics of finite order on a holomorphic vector bundle by imposing estimates on the holomorphic bisectional curvature. We generalize the vanishing theorem of Griffiths and Cornalba regarding Hermitian bundles of finite order to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of a vector bundle to projective space. We show that the projectivization of a Finsler bundle of finite order can be immersed into a projective space of sufficiently large dimension via a map of finite order.