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INTERSECTION NUMBER AND SOME METRICS ON TEICHMÜLLER SPACE

Sun, Zongliang 大阪大学 DOI:10.18910/73741

2020.01

概要

be a closed surface of genus g ≥ 2, T (X) the Teichm¨uller space of X, and C(X) the space
of geodesic currents on X. Let i : C(X) × C(X) → R be the intersection number between
geodesic currents, which is the generalization of geometric intersection number between
(homotopy classes of) closed curves. It turns out that (see, e.g. [4, 5, 11]) much information
about C(X), such as the topology of C(X), is governed by the intersection number. In [4],
Bonahon established an embedding L : T (X) → C(X) of the Teichm¨uller space into C(X),
which maps a hyperbolic metric to the corresponding Liouville current. Moreover, Bonahon
[4] was able to rebuild the Weil-Petersson metric and Thurston’s compactification of the
Teichm¨uller space with the aid of the intersection number between geodesic currents. In this
paper, we make an attempt to study some aspects of the Teichm¨uller space by the intersection
number, via Bonahon’s embedding. We are mainly concerned with quantitative comparisons
and topological behaviors of the intersection number and some metrics on T (X).
There are many interesting metrics on T (X), among them we have the Teichm¨uller metric
dT , the length spectrum metric dL and Thurston’s asymmetric metrics dPi , i = 1, 2. These
metrics have been studied by many authors [1, 2, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17]. As our
first result, we describe some quantitative comparisons between the intersection number and
these metrics as follows.
Throughout the paper, for notational convenience, we will frequently denote L(ρ) by Lρ
for ρ ∈ T (X). ...

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参考文献

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Intersection Number and Metrics

149

Zongliang Sun

Department of Mathematics

Shenzhen University

Shenzhen 518060

P. R. China

Hui Guo

Department of Mathematics

Shenzhen University

Shenzhen 518060

P. R. China

e-mail: hguo@szu.edu.cn

...

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