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We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum dominates another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface.
Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular nei
We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary co
Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we
In this paper, $S$ denotes a hyperbolic surface homeomorphic to a punctured torus or a pairs of pants. Our interest is the study of emph{textbf{combinatorial $k$-systoles}} that is geodesics with self-intersection number greater than $k$ and with min
This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinato