While the equivalence between ampleness and positivity holds for vector bundles of rank one, its higher rank counterpart known as Griffith's conjecture is still open. There is also a similar but weaker conjecture by Kobayashi who proposed to use Finsler rather than Hermitian metrics to study the equivalence. We will review these two conjectures and state our progress. One of our results is that we can construct a positively curved Finsler metric on if the symmetric power of the dual  has a certain negatively curved Hermitian metric.