  In this paper, we extend the construction of pressure metrics to
Teichm\"uller spaces of surfaces with punctures. This construction recovers
Thurston's Riemannian metric on Teichm\"uller spaces. Moreover, we prove the
real analyticity and the convexity of Manhattan curves of the 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.
