Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

Abstract

In this paper, we extend the construction of pressure metrics to Teichmuller spaces of surfaces with punctures. This construction recovers Thurston’s Riemannian metric on Teichmuller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.