the ‘Ricci-flow method’.

A more straightforward approach to solve an elliptic PDE is to iteratively linearize it, i.e. apply Newton’s method. The point here is that there exist direct methods to solve the resulting linear PDE. For example, using real-space finite differencing this PDE appears as a sparse system of linear equations, for which fast solvers exist such as the biconjugate gradient method. Such solvers are not affected by negative modes (although they are by zero modes). Whilst more complicated to implement in practice, Newton’s method avoids having to tune families of initial data—just one guess metric in the basin of attraction of the desired solution is sufficient. Unlike the Ricci-flow method, the trajectory of Newton’s method in the space of geometries does depend on the reference connection, and hence so does the basin of attraction of a solution. In principle this makes the method somewhat less elegant than Ricci-flow, although in practice it appears to make little difference.

In summary, finding Euclidean Ricci-flat geometries and static vacuum Lorentzian solutions, including black holes, can be phrased as the same problem and solved using the same unified, covariant elliptic approach. The problem of dealing with Euclidean negative modes in either context 