Harmonic function on a Riemann surface

A real-valued function on a Riemann surface is harmonic when its expression in every holomorphic coordinate chart has vanishing planar Laplacian.

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- Conformal invariance of harmonicity Harmonic function on a Riemann surface

Conformal invariance of harmonicity

Under a holomorphic coordinate change

w = w (z)

,

Δ_{z} (H \circ w) = ∣ w^{'} (z) ∣^{2} (Δ_{w} H) \circ w,

so the condition

Δ H = 0

is independent of the holomorphic chart.

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