An ancient solution of the Ricci flow in dimension 4 converging to the Euclidean Schwarzschild metric (Q2252676)

Consider the Euclidean Schwarzschild metric \[ g_0 = (1-r^{-1})dt^2 + (1-r^{-1})^{-1}dr^2 + r^2 d\Omega^2 \] on \(M = [0,4\pi] \times (1,\infty) \times S^2\), where \(d\Omega^2\) is the standard metric of the sphere. It is obtained from the Schwarzschild metric by changing the time coordinate to \(\tau = -it\) and taking black-hole mass to \(1/2\). It is Ricci flat, in particular a critical point of the Einstein-Hilbert functional \(S(g) = -\int R \sqrt{g} dt\, dr\, d\Omega\). The author proves that this critical point is unstable. It is derived from the main result of the paper that \(g_0\) is the limit of an ancient solution of the Ricci flow at \(t = - \infty\). More precisely, the author proves: Theorem. There exists \(N \in \mathbb{R}\) and an ancient solution \(g(x,t)\) of the Ricci flow \[ \partial_t g(x,t)= - 2\mathrm{Ric}(x,t)\quad \forall x \in M,\, t \in (-\infty,N), \] such that \(g(x,t)\to g_0\) as \(t \to \infty\).