On the Convexity of the Kobayashi Metric on a Taut Complex Manifold

Let M be an m-dimensional complex manifold. We recall the definition of the Kobayashi-Royden pseudo-metric on M : FM (ξ) = inf{t > 0| ∃f ∈ O(∆,M) such that tf∗(d/dζ|ζ=0) = ξ}, (1.1) where ξ ∈ T pM is a holomorphic tangent vector, ∆ = {ζ ∈ C| |ζ| < 1}, and O(∆,M) = {f : ∆ → M | f is a holomorphic mapping}. Then, FM has the following properties: (i) FM (ξ) ≥ 0 for any ξ ∈ T pM ; (ii) FM (λξ) = |λ|FM (ξ) for any λ ∈ C; (iii) FM is upper semi-continuous on the holomorphic tangent bundle T pM , moreover if M is taut, that is, O(∆,M) is a normal family; (iv) FM is continuous on TM ; (v) FM (ξ) = 0 if and only if ξ = 0. Hence we see that FM is a metric on M , if M is taut. Let v ∈ TpM be a real tangent vector. We can uniquely write v = ξ + ξ̄ with ξ ∈ T pM . We set FM (v) = 2FM (ξ). Then FM induce a pseudodistance dM on M as follows: