Recall that Kobayashi metric is defined on any complex manifold $M$. This is a pseudo-metric according to which a tangent vector $v$ at $P$ has length at most $1$ if there is holomorphic map from the open unit disk in $\mathbb C$ to the manifold that sends $0$ to $P$ and $\frac{\partial}{\partial x}$ to $v$.

http://en.wikipedia.org/wiki/Kobayashi_metric

I would like to know if the following analogue of such pseudo-metric makes sense.

**Definition.** Let $M$ be a Riemannian manifold and let us say that $v$ has length at most $1$ (for the new pseudo-metric) if there is a conformal minimal immersion of the unite disk to $M$ that sends a unite vector at the centre of the disk to $v$.

**Question.** Are there many examples for which this pseudo-metric does not vanish? Was such a definition studied by someone?

**Remark.** Clearly in the case $M$ is a Riemann surface this construction gives us the usual Kobayashi metric (i.e it is trivial for $\mathbb C^1$, $\mathbb CP^1$, $T^2$, $\mathbb C^*$ and is a metric of constant negative curvature otherwise).

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