The computation of conformal map by harmonic diffeomorphisms between surfaces

Harmonic map is the critical point of the corresponding integral with respect to the square norm of the gradient or energy density. The harmonic energy defined on Riemann surfaces will decrease along its gradient line direction and reduce to the limit conformal map. Harmonic map between surfaces is a diffeomorphism which is associated to a unique Beltrami differential. So a harmonic diffeomorphism sequence corresponds to an Beltrami differential sequence. When boundary map is restricted on a unit circle, the Beltrami differential sequence changes with constant conformal modulus. In this paper, we consider the conformal map between surfaces by the Beltrami differential sequence with constant conformal modulus, which is equivalent to a decreasing harmonic energy sequence with fixed boundary correspondence, and provide the corresponding algorithms for numerical computation. Furthermore, we will discuss the convergence of proposed algorithms, which provides theoretical foundation for numerical experiments.