Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 118 / 2 / b / Solution

The real coordinate functions are real and imaginary parts of the holomorphic coordinates

z_{j}

. Part (a), together with the real coefficients of the Laplace-Beltrami operator, therefore shows that every

x_{l}

is a harmonic function. For the given divergence-of-gradient convention, applying the operator to

x_{l}

gives

0 = Δ_{d} x_{l} = \frac{1}{d e t g} \sum_{k = 1}^{2 n} \frac{\partial}{\partial x _{k}} (det g g^{k l}) .

These are the harmonic coordinate identities. Expanding the operator on an arbitrary smooth function now gives

Δ_{d} u = \sum_{k, l = 1}^{2 n} g^{k l} \frac{\partial ^{2} u}{\partial x _{k} \partial x _{l}} + \sum_{l = 1}^{2 n} [\frac{1}{d e t g} \sum_{k = 1}^{2 n} \frac{\partial}{\partial x _{k}} (det g g^{k l})] \frac{\partial u}{\partial x _{l}} .

The bracketed coefficients vanish. Hence there is no first-derivative term in these coordinates:

Δ_{d} u = k, l = 1 \sum 2 n g^{k l} \frac{\partial ^{2} u}{\partial x _{k} \partial x _{l}} .

The reason is that the real components of holomorphic coordinates are harmonic coordinates for a Kähler metric; a general real coordinate change need not retain this simplification.

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