The p-Weil–Petersson Teichmüller Space and the Quasiconformal Extension of Curves
Huaying Wei, Katsuhiko Matsuzaki
Abstract
Abstract We consider the correspondence between the space of p -Weil–Petersson curves $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>γ</mml:mi></mml:math> on the plane and the p -Besov space of $$u=\log \gamma '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mo>log</mml:mo><mml:msup><mml:mi>γ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math> on the real line for $$p > 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> . We prove that the variant of the Beurling–Ahlfors extension defined by using the heat kernel yields a holomorphic map for u on a domain of the p -Besov space to the space of p -integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichmüller projection from the space of p -integrable Beltrami coefficients to the p -Weil–Petersson Teichmüller space.