Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?
2026-03-30 14:27:16.1774880836
Is conformal equivalence the same as topological equivalence?
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The simplest nontrivial case is an open cylinder. Each of these is conformally equivalent to a circular annulus $\{z: r<|z|<R\}$, and two such annuli are conformally equivalent iff they have the same ratio $R/r$. In other words, annuli are classified by one real parameter, conformal modulus.
The case of torus was discussed in conformally equivalent flat tori.
In general, the key term is the moduli space of a surface. As Alan noted in comments, its dimension can be expressed in terms of the genus and the number of boundary components. The formidable [to me] Wikipedia article suggests that the subject is not an easy one to enter, while the MathOverflow discussion Intuition behind moduli space of curves attempts to give some low-tech insights.