A bordered Riemann surface X can be defined as a region on a compact Riemann surface whose boundary ∂X consists of finitely many disjoint Jordan curves.
\ell(\gamma)=\int_\gamma \rho(z) \, |dz|,\quad A(D)=\int_D\rho^2(x+iy) \, dx \, dy, \quad z=x+iy.