a counterexample by taking any compact region of any of the known non-compact steady Ricci solitons.) The theorem also covers the case with negative cosmological constant: $R_{\mu\nu} - \nabla_{(\mu}\xi_{\nu)} = \kappa g_{\mu\nu}$, with $\kappa$ a negative constant, implies $R_{\mu\nu} = \kappa g_{\mu\nu}$. On the other hand, for positive $\kappa$ there are known counterexamples (called *shrinking Ricci solitons*), for example on the first and second del Pezzo surfaces [@MR1145263; @MR2084775; @MR2243675; @HeadrickdP2]. However, the existence of such metrics should not present a major problem as long as they are isolated, since it is easy enough to check whether one’s algorithm is converging to a true solution to the Einstein equation or merely to a soliton, and in the latter case to restart it with a different initial guess. (Similar comments presumably apply in the presence of matter fields, but we have not studied this case in detail.)

Let us now discuss what boundary conditions are appropriate for equation . It is well known that the Einstein equation, being effectively $D(D-1)/2$ equations for the same number of variables, requires the same number of boundary conditions. For example, one can fix the induced metric on the boundary or its extrinsic curvature, which are 