On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type

<p>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.</p> <p>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 ...