Lines of Minima and Teichmüller Geodesics
Lines of Minima and Teichmüller Geodesics
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极小值线和 Teichmüller 测地线
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
C. Series
中科院分区：
文献类型：
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作者：
Young;Kasra Rafi;C. Series
Abstract.For two measured laminations ν+ and ν− that fill up a hyperbolizable surface S and for $$t,in,(-infty,infty)$$, let $${mathcal{L}}_t$$ be the unique hyperbolic surface that minimizes the length function etl(ν+) + e-tl(ν−) on Teichmüller space. We characterize the curves that are short in $${mathcal{L}}_t$$ and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface $${mathcal{G}}_t$$ on the Teichmüller geodesic whose horizontal and vertical foliations are respectively, etν+ and etν−. By deriving additional information about the twists of ν+ and ν− around the short curves, we estimate the Teichmüller distance between $${mathcal{L}}_t$$ and $${mathcal{G}}_t$$. 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 bounded independently of t.
Abstract.For two measured laminations ν+ and ν− that fill up a hyperbolizable surface S and for $$t,in,(-infty,infty)$$, let $${mathcal{L}}_t$$ be the unique hyperbolic surface that minimizes the length function etl(ν+) + e-tl(ν−) on Teichmüller space. We characterize the curves that are short in $${mathcal{L}}_t$$ and estimate their lengths. We find that the short curves coincide with the curves that are short in the surface $${mathcal{G}}_t$$ on the Teichmüller geodesic whose horizontal and vertical foliations are respectively, etν+ and etν−. By deriving additional information about the twists of ν+ and ν− around the short curves, we estimate the Teichmüller distance between $${mathcal{L}}_t$$ and $${mathcal{G}}_t$$. 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 bounded independently of t.