Harmonic coordinates for Ricci flow

Question

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.

Answer 1

There are at least two problems if you naively approach the problem from that point of view:

1. How do you propose to time-evolve the harmonic coordinate system? For the Einstein's equation the coordinate system is more properly called "wave coordinates" as the coordinate functions solve the wave equation (compare to the Riemannian case when the coordinate functions solve the Laplace equation).
2. Harmonic coordinate systems are not necessarily global. For hyperbolic equations like Einstein's equation, finite speed of propagation implies that you can cut up into small neighborhoods, and solve in the domain of dependence o that neighborhood in the (local) wave coordinate system. For parabolic equation one cannot use the same method.

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.