Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Finsler stucture of the Teichm\"uuller metric and Thurston's
  asymmetric metric   We highlight several analogies between the Finsler (infinitesimal) properties
of Teichm\"uller's metric and Thurston's asymmetric metric on Teichm\"uller
space. Thurston defined his asymmetric metric in analogy with Teichm\"ullers'
metric, as a solution to an extremal problem, which consists, in the case of
the asymmetric metric, of finding the best Lipschitz maps in the hoomotopy
class of homeomorphisms between two hyperbolic surface. (In the Teichm\"uller
metric case, one searches for the best quasiconformal map between two conformal
surfaces.) It turns out also that some properties of Thurston's asymmetric
metric can be used to get new insight into Teichm\"uller's metric. In this
direction, in analogy with Thurston's formula for the Finsler norm of a vector
for the asymmetric metric that uses the hyperbolic length function, we give a
new formula for the Finsler norm of a vector for the Teichm\"uller metric that
uses the extremal length function. We also describe an embedding of projective
measured foliation space in the cotangent space to Teichm\"uller space whose
image is the boundary of the dual of the unit ball in the tangent space
representing vectors of norm one for the Finsler structure associated to the
Teichm\"uller metric.

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The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-th
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex