  We study growth of holomorphic vector bundles E over smooth affine manifolds.
We define Finsler metrics of finite order on E by estimates on the holomorphic
bisectional curvature. These estimates are very similar to the ones used by
Griffiths and Cornalba to define Hermitian metrics of finite order. We then
generalize the Vanishing Theorem of Griffiths and Cornalba to the Finsler
context. We develop a value distribution theory for holomorphic maps from the
projectivization of E to projective space. We show that the projectivization of
E can be immersed into a projective space of sufficiently large dimension via a
map of finite order.
