Conformal invariance of harmonicity
= Conformal invariance of harmonicity
Under a holomorphic coordinate change $w=w(z)$,
$$
\Delta_z(H\circ w)=|w'(z)|^2(\Delta_wH)\circ w,
$$
so the condition $\Delta H=0$ is independent of the holomorphic chart.
= Conformal invariance of harmonicity
Under a holomorphic coordinate change $w=w(z)$,
$$
\Delta_z(H\circ w)=|w'(z)|^2(\Delta_wH)\circ w,
$$
so the condition $\Delta H=0$ is independent of the holomorphic chart.