  In this paper, we define a new metric structure on the shape space of a high
genus surface. We introduce a rigorous definition of a shape of a surface and
construct a metric based on two energies measuring the area distortion and the
angle distortion of a quasiconformal homeomorphism. We show that the energy
minimizer in a fixed homotopy class is achieved by a quasiconformal
homeomorphism by the lower semicontinuity property of these two energies.
