Proof of the Lemma

Since f+ig is complex analytic, f and g obey the Cauchy-Riemann equations, f_u=g_v f_v=-g_u. Thus for the parametrization x(u,v)=(u,v,f(u,v),g(u,v)) we get

E = G = 1+ f_u²+ f_v²  and  F = 0.

(so, x is conformal). Rewrite the above for machine input and apply tensorK. The result is lengthy. But the Cauchy-Riemann equations imply that f is harmonic, that is, f_uu+f_vv=0. (Pf. f_uu=g _vu=g_uv= -f_vv.) Derivatives give two more relations, so substitute

f_vv -f_uu ,   f_uvv -f_uuu,  f_vvv -f_uuv

into the previous curvature result. Machine-simplification then gives K as required.   QED

For practice, generalize to the case of two complex analytic functions, f₁+ig₁, f₂+ig₂.

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