uniqueness of solutions to . Under a small diffeomorphism, generated by a vector field $w^\mu$, the change in $\xi^\mu$ is $\delta\xi^\mu = -\Delta_Vw^\mu$, where $$\label{deltaxi}
\Delta_Vw^\mu \equiv -\frac12\nabla^2w^\mu - \frac12{R^\mu}_\nu w^\nu + (\Gamma^\mu_{\lambda\nu} - \tilde\Gamma^\mu_{\lambda\nu})\nabla^\lambda w^\nu\,.$$ $\Delta_V$ is a strictly elliptic operator, so locally $\xi^\mu + \delta\xi^\mu = 0$ will have a unique solution for $w^\mu$. The global existence and uniqueness problems are more difficult, of course. On a compact manifold the operator defined by will have at most a finite-dimensional kernel, so the space of harmonic maps is at most finite-dimensional (see also [@EellsLemaire]). As far as existence is concerned, there is no general proof, but there are many partial results (including for the case of manifolds with boundary); for example, Eells and Sampson [@EellsSampson] proved existence under the assumption of non-positive sectional curvatures for $\tilde g_{\mu\nu}$. As far as we are aware, there are no known counterexamples.[^5]

A posteriori gauge fixing
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We now turn to the problem of implementing our gauge-fixing condition $\xi^\mu = 0$. This is a differential equation for the metric, in contrast to algebraic gauge-fixing conditions such as fixing the values of certain metric components. In the latter case, when implementing a relaxation method, 