Bounded combinatorics and the Lipschitz metric on Teichmüller space.
(With Anna Lenzhen and Jing Tao)
To appear: Geometriae Dedicata.
Considering the Teichmüller space of a surface equipped with Thurston’s
Lipschitz metric, we study geodesic segments whose endpoints have bounded
combinatorics. We show that these geodesics are cobounded, and that the closest
point projection to these geodesics is strongly contracting. Consequently, these
geodesics are stable. Our main tool is to show that one can get a good estimate
for the Lipschitz distance by considering the length ratio of ﬁnitely many curves.
Length of a curve is quasi-convex along a Teichmüller geodesic.
(With Anna Lenzhen)
J. Differential Geom. 88 (2011), no. 2, 267–295.
We show that for every simple closed curve α, the extremal length and the
hyperbolic length of α are quasi-convex functions along any
Teichmüller geodesic. As a corollary, we conclude that, in Teichmüller
space equipped with the Teichmüller metric, balls are quasi-convex.
Curve complexes with connected boundary are rigid.
(With Saul Schleimer)
Duke Math. J. Volume 158, Number 2 (2011), 225-246.
When the boundary of the curve complex is connected any quasi-isometry
is bounded distance from a simplicial automorphism. As a consequence,
when the boundary is connected the quasi-isometry type of the curve
complex determines the homeomorphism type of the surface.
Grafting rays fellow travel Teichmüller geodesics.
(With Young-Eun Choi and David Dumas)
International Mathematics Research Notices (2011); doi: 10.1093/imrn/rnr104
Given a measured geodesic lamination L on a hyperbolic surface S,
grafting the surface along sL (s > 0) defines a one-parameter family
gr( sL, S) of conformal structures in the Teichmüller space,
called the grafting ray. We show that every grafting ray, after
reparametrization, is a Teichmüller quasi-geodesic and stays in
a bounded neighborhood of a Teichmüller geodesic. As part of
our approach, we show that grafting rays have controlled dependence
on the base. That is, there exists a constant K such that for
Riemann surfaces R and S, the distance between gr(sL, R) and gr(sL,
S) is less than K time the distance between R and S.
Length spectra and degeneration of flat metrics
(With Moon Duchin and Christopher Leininger)
Invent. Math. 182 (2010), no. 2, 231–277.
In this paper we consider ﬂat metrics (semi-translation structures)
on surfaces of ﬁnite type. There are two main results. The ﬁrst is a complete
description of when a set of simple closed curves is spectrally rigid, that is,
when the length vector determines a metric among the class of ﬂat metrics.
Secondly, we give an embedding into the space of geodesic currents and use this
to get a boundary for the space of ﬂat metrics. The geometric interpretation
is that ﬂat metrics degenerate to mixed structures on the surface: part ﬂat
metric and part measured foliation.
Divergence rate of geodesics in Teichmüller space and mapping
(with Moon Duchin)
Geom. Funct. Anal. 19 (2009), no. 3, 722--742.
We say a function f(R) is a divergence function for two geodesic
rays in a metric space that share a basepoint if points on these
rays that are distance R from the basepoint can be connected along a
path that remain distance at least R from the basepoint and has
length less than f(R). We show that every two geodesic rays in the
Teichmüller space that share a basepoint have a quadratic
divergence function. Furthermore, we show that this esmiate is
sharp by providing examples where every divergence function is at
least quadratic. The same is also true for geodesic rays in the
mapping class group.
Covers and the curve complex.
(With Saul Schleimer)
Geom. Topol. 13 (2009), no. 4, 2141--2162.
We provide the first non-trivial examples of quasi-isometric
embeddings between curve complexes. These are induced either by
puncturing a closed surface or via finite-sheeted coverings. As a
corollary, we give new quasi-isometric embeddings between mapping
Lines of minima and Teichmüller geodesics.
(With Young-Eun Choi and Caroline Series)
Geom. Funct. Anal. 18 (2008), no. 3, 698--754.
We characterize the curves that are short along a line of minima in
Teichmüller space and estimate their lengths. We find that the
short curves coincide with the curves that are short along the
corresponding Teichmüller geodesic. By deriving additional
information about the twisting parameters around the short curves,
we estimate the Teichmüller distance between the line of minima
and the Teichmüller geodesic. We deduce that this distance can
be arbitrarily large, but that if S is a once-punctured torus or
four-times-punctured sphere, the distance is uniformly bounded.
Lines of minima are uniformly quasigeodesic.
(With Young-Eun Choi and Caroline Series)
Pacific J. Math. 237 (2008), no. 1, 21-44.
We use the results in "Lines of minima and Teichmüller
geodesics" and "A combinatorial model for the Teichmüller
metric" to prove that every line of minima is a quasi-geodesics in
the Teichmüller metric.
Comparison between Teichmüller and Lipschitz metrics.
(With Young-Eun Choi)
J. Lond. Math. Soc. (2) 76 (2007), no. 3, 739-756.
We study the Lipschitz metric on Teichmüller space (defined by
Thurston) and compare it with the Teichmüller metric. We show
that in the thin part of Teichmüller space the Lipschitz metric
is approximated up to bounded additive distortion by the sup metric
on a product of lower-dimensional spaces (similar to the
Teichmüller metric as shown by Minsky), and in the thick part,
the two metrics are comparable within additive error. However, these
metrics are not comparable in general; we construct a sequence of
pairs of points in Teichmüller space whose distances approach
zero in the Lipschitz metric while they approach infinity in the
A combinatorial model for the Teichmüller metric.
Geom. Funct. Anal. 17 (2007), no. 3, 936-959.
We study how the length and the twisting parameter of a curve change
along a Teichmüller geodesic. We then use our results to
provide a formula for the Teichmüller distance between two
hyperbolic metrics on a surface, in terms of the combinatorial
complexity of curves of bounded lengths in these two metrics.
Thick-thin decomposition for quadratic differentials.
Math. Res. Lett. 14 (2007), no 2, 333-341.
We provide a concrete description of the geometry of a quadratic
differential metric; the surface is decomposed into thick pieces
that are glued along flat annuli. Every thick piece Y of S has an
scaling factor such that, for every essential curve in Y , the
quadratic differential length of this curve is equal to the scaling
factor times its hyperbolic length (up to multiplicative constants
depending only on the topology of S).
A charaterization of short curves of a geodesic in Teichmüller
Geom. Topol. 9 (2005) 179-202.
We provide a combinatorial condition characterizing curves that are
short along a Teichmüller geodesic. This condition is closely
related to the condition provided by Minsky for curves in a
hyperbolic 3-manifold to be short. We show that short curves in a
hyperbolic manifold homeomorphic to SxR are also short in the
corresponding Teichmüller geodesic, and we provide examples
demonstrating that the converse is not true.