In this thesis, we study the geometry of Teichmuller space of punctured Riemann surfaces.

We use L2 Hodge theory to describe the deformation theory for punctured Riemann surfaces,

in which we defined Weil-Petersson metric, Hodge metric and Kodaira-Spencer map. We

also give a new proof of Wolpert's curvature formula by computing the expansion of volume

form and the Kodaira-Spencer map. We use Wolpert's formula to estimate upper bound for

various curvature tensor. We construct an extension of pluricanonical form and compare it

to the expansion of the Kodaira-Spencer map under Hodge metric.