How to determine all the complex structures on torus $T^2$?

Question

I have known that the lattice given by the pair $(\tau_1,\tau_2)$ can determine a complex structures on torus $T^2$.

But how to prove that all the complex structures of torus can be obtained in this way?

Answer 1

This follows from the Riemann mapping theorem.

If you consider a compatible Riemannian metric on the torus, the claim can be expressed as follows: every metric on the torus has a flat metric in its conformal class.