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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
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Bruce Kleiner has a unpublished note where he used this construction (Brady-like hyperbolicity) to characterize closed Riemannian manifolds with word-hyperbolic fundamental groups.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). –  user46438 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

1 Answer 1

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.

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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

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