Publishing type：Research paper (scientific journal)   Publisher：Project Euclid  

DOI： 10.3792/pjaa.100.013

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Language：English   Publisher：Department of Mathematics, Tokyo Institute of Technology  

For a generalized Cantor set <i>E</i>(<i>ω</i>) with respect to a sequence <img align="middle" src="./Graphics/abst-1.jpg"/>, we consider Riemann surface <img align="middle" src="./Graphics/abst-2.jpg"/> and metrics on Teichmüller space <i>T</i>(<i>X</i>_{<i>E</i>(<i>ω</i>)}) of <i>X</i>_{<i>E</i>(<i>ω</i>)}. If <i>E</i>(<i>ω</i>) = <img align="middle" src="./Graphics/abst-3.jpg"/> (the middle one-third Cantor set), we find that on <img align="middle" src="./Graphics/abst-4.jpg"/>, Teichmüller metric <i>d</i>_<i>T</i> defines the same topology as that of the length spectrum metric <i>d</i>_<i>L</i>. Also, we can easily check that <i>d</i>_<i>T</i> does not define the same topology as that of <i>d</i>_<i>L</i> on <i>T</i>(<i>X</i>_{<i>E</i>(<i>ω</i>)}) if sup <i>q</i>_<i>n</i> = 1. On the other hand, it is not easy to judge whether the metrics define the same topology or not if inf <i>q</i>_<i>n</i> = 0. In this paper, we show that the two metrics define different topologies on <i>T</i>(<i>X</i>_{<i>E</i>(<i>ω</i>)}) for some <img align="middle" src="./Graphics/abst-5.jpg"/> such that inf <i>q</i>_<i>n</i> = 0.

DOI： 10.2996/kmj47103

DOI： 10.48550/arxiv.2211.04897

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Publisher：American Mathematical Society (AMS)  

On the Teichmüller space <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper R 0 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">T(R_0)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of a hyperbolic Riemann surface <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R 0">
<mml:semantics>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">R_0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, we consider the length spectrum metric <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript upper L">
<mml:semantics>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">d_L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R 0">
<mml:semantics>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">R_0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is of finite type, then <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript upper L">
<mml:semantics>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">d_L</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> defines the same topology as that of Teichmüller metric <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript upper T">
<mml:semantics>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>T</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">d_T</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper R 0 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">T(R_0)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. In 2003, H. Shiga extended the discussion to the Teichmüller spaces of Riemann surfaces of infinite type and proved that the two metrics define the same topology on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis upper R 0 right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">T(R_0)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R 0">
<mml:semantics>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">R_0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> satisfies some geometric condition. After that, Alessandrini-Liu-Papadopoulos-Su proved that for the Riemann surface satisfying Shiga’s condition, the identity map between the two metric spaces is locally bi-Lipschitz.



In this paper, we extend their results; that is, we show that if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R 0">
<mml:semantics>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:annotation encoding="application/x-tex">R_0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> has bounded geometry, then the identity map <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T left-parenthesis upper R 0 right-parenthesis comma d Subscript upper L Baseline right-parenthesis right-arrow left-parenthesis upper T left-parenthesis upper R 0 right-parenthesis comma d Subscript upper T Baseline right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>T</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>L</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo stretchy="false">→<!-- → --></mml:mo>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>T</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>d</mml:mi>
<mml:mi>T</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">(T(R_0),d_L) \to (T(R_0),d_T)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is locally bi-Lipschitz.

DOI： 10.1090/ecgd/316

DOI： 10.2996/kmj47103_references_DOI_X23C6T8ShAU2F19MhqGjlpPISpd

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Publisher：Finnish Mathematical Society  

We consider the length spectrum metric dL in infinite dimensional Teichmuller space T(R0). It is known that dL defines the same topology as that of the Teichmuller metric dT on T(R0) if R0 is a topologically finite Riemann surface. In 2003, Shiga proved that dL and dT define the same topology on T(R0) if R0 is a topologically infinite Riemann surface which can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded by some positive constants from above and below. In this paper, we extend Shiga's result to Teichmuller spaces of Riemann surfaces satisfying a certain geometric condition.

DOI： 10.5186/aasfm.2014.3925

DOI： 10.2996/kmj47103_references_DOI_OKdZk8UcNpC8HendkgJSkxDCYpK

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Publishing type：Research paper (scientific journal)   Publisher：Tokyo Institute of Technology, Department of Mathematics  

DOI： 10.2996/kmj/1309829545

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