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-Ampère type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kähler geometry, related to the construction of flat maps for the Mabuchi metric.

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We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains. Lemma 2.1. If M is a simply connected Riemann surface and PIf dim V =1, we recover the traditional result by taking logarithms.Proof. This lemma is actually true for M a simply connected complex manifold, see [Dem97, Chapter V.6]. Although the bundle is of finite rank there, the proof carries over to infinite rank easily. Lemma 2.2. Let M be a compact Riemann surface with boundary, PProof. It is basically the same as the proof of [Lem17, Corollary 3.3].

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