# Harmonic coordinates for Ricci flow

It is customary to use DeTurck's argument (or Hamilton's original one involving the Nash-Moser iteration) for proving local existence of the Ricci flow. I am wondering why one cannot use harmonic coordinates for this purpose, as can be done for the Einstein equations.

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That said, the DeTurck trick can be interpreted as the "harmonic coordinate" version of the proof. In the DeTurck trick you solve the modified Ricci flow with a vector field $X^a = g^{bc}\Gamma_{bc}^a$ where the $\Gamma$ are the analogues of the Christoffel symbol "relative to a fixed background metric". Now, the choice of harmonic coordinate system is one in which $0 = g^{bc}\Gamma_{bc}^a$ (the real Christoffel symbol relative to the coordinates now). So we can see DeTurck's trick as circumventing the difficulty in choosing a global, truly harmonic coordinate system, by compensating it with a time-dependent diffeomorphism that "gets rid of" that extra non-harmonicity.
The correct analogue for the DeTurck trick in Einstein's equations is not the simple o' wave coordinate system. Instead, it is what Choquet-Bruhat calls the "$\hat{e}$ wave gauge" and what is sometimes known as the "wave-map gauge"; for more details you should consult chapter VI.7 of her recent monograph General Relativity and the Einstein Equations.
There's actually not much difference between the homogeneous and inhomogeneous version. The power of the DeTurck trick (and of the various wave gauges) is that the equations in that gauge becomes manifestly hyperbolic with a diagonal principal symbol, that is, the Ricci curvature term(s) can be replaced by (roughly speaking) $g^{ij}\partial^2_{ij}g_{kl}$. Now since Einstein's equation can always be written as $R_{ij} = T_{ij} - \frac12 T g_{ij}$ (moving the trace term to the other side) the principal part is purely determined by Ricci. – Willie Wong Jul 24 '12 at 19:22