  Following Kobayashi, we consider Griffiths negative complex Finsler bundles,
naturally leading us to introduce Griffiths extremal Finsler metrics. As we
point out, this notion is closely related to the theory of interpolation of
norms, and is characterized by an equation of complex Monge--Amp\`ere type,
whose corresponding Dirichlet problem we solve. As applications, we prove that
Griffiths extremal Finsler metrics quantize solutions to a natural PDE in
K\"ahler geometry, related to the construction of flat maps for the Mabuchi
metric.
