From the Encyclopedia of Mathematics website, I read that constant mean curvature (CMC) surfaces, in particular minimal surfaces, are isothermal surfaces, i.e. a surface whose curvature lines form an isothermal net. But how can I prove this?
Edit: From the comment by @WillJagy, I see from the Weierstrass–Enneper parameterization, curvature lines of minimal surfaces are isothermal coordinates. But what about a general CMC surface?