Dynamical degrees of freedom for higher genus Riemann surface in (2+1)-dimensional general relativity

A homogeneous two-dimensional metric including the degrees of freedom of Teichm"uller deformation is developed. The Teichm"uller deformation is incorporated by affine stretching of complex structure. According to Yamada’s investigation by pinching parameter, concrete formulation for a higher genus Riemann surface can be realized. We will have a homogeneous standard metric including the dynamical degrees of freedom as Teichm"uller deformation in a leading order of the pinching parameter, which would be treated as homogeneous anisotropic metric for a double torus universe, which satisfy momentum constraints.