# A “Riemannian” analogue of Kobayashi metric?

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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It seems to me that if the manifold is not conformally flat, then there might not be any conformal maps of the unit disk into a neighborhood of a point. If it's conformally flat, then presumably you get something similar to what happens for Riemann surfaces. That might still be interesting to study. –  Deane Yang May 14 '13 at 23:57
Bruce Kleiner has a unpublished note where he used this construction (Brady-like hyperbolicity) to characterize closed Riemannian manifolds with word-hyperbolic fundamental groups. Gabai and Kazez has a published paper “Group Negative Curvature for 3-Manifolds with Genuine Laminations”, Geom. and Top., 2 (1998) 65-77, where they worked out the case of 3-dimensional targets. Mosher and Oertel earlier had a combinatorial version. (Mosher will probably make further comments here.) –  Misha May 15 '13 at 3:03
Misha, thanks for your comment I'll check the paper. –  aglearner May 15 '13 at 11:58
The same notion of a "real Kobayashi metric" is described by Gromov in his "Metric structures.." book (see Page 8). –  harish Feb 3 at 8:09
@Misha: it seems to me that your comment should be made into an answer so that it could be accepted. –  Benoît Kloeckner Feb 3 at 13:10
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This metric is already interesting the case of a domain $U$ in the Riemann sphere, if we replace conformal maps $f : \Delta \rightarrow U$ with Moebius transformations. Then maximal disks in $U$ correspond to supporting hyperplanes for the boundary of the convex hull of the boundary of $U$ in hyperbolic 3-space. The metric itself is obtained from the boundary of the convex hull -- which is intrinsically a hyperbolic surface, except in degenerate cases -- by grafting along the bending lamination. (The metric decomposes into pieces of constant negative curvature -- where the supporting hyperplane is unique -- and into flat pieces -- where it is not.) This metric is part of Thurston's approach to complex projective structures (on any Riemann surface), see S. Matsumoto, "Foundations of flat conformal structure". A similar story takes place in higher dimensions, where it is no longer necessary to require $f$ is a Moebius map --- this is automatic.
Dear Curtis, thank you for this remark! Have I got you correctly, that you put $U$ inside $\partial H^3$. Do you know if someone studied the definition that I propose? –  aglearner May 20 '13 at 14:43
This approach of Thurston is similar to Kobayashi's approach in Projective structures with trivial intrinsic pseudodistance'', 1978, maybe earlier than Thurston? He was already working with projective connections, but in the real category rather than complex, defining a Kobayashi pseudometric. –  Ben McKay May 20 '13 at 16:31