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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.
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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 neighborhood of $n+2$ generic hyperplanes from $mathbb{CP}^n$. As is well-known, every compact surface of genus $ggeqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a 4-manifold that decomposes into pairs of pants.
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 components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.
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 call the Fenchel-Nielsen Teichmuller space associated to the pair $(mathcal{P},X)$. This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers cite{ALPSS}, cite{various} and cite{local}, and we compared it to the classical Teichmuller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.
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 minimal combinatorial length. We show that the maximal intersection number $I_k$ of combinatorial $k$-systoles grows like $k$ and $underset{krightarrow+infty}{limsup}(I_k(S)-k)=+infty$. This answer -- in the case of a pairs of pants and a punctured torus -- a weak version of Erlandsson-Palier conjecture, originally stated for the geometric length.
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 combinatorial Yamabe flow is a gradient flow of a concave function. The long time behavior of the flow and the geometric meaning is investigated.
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