The evolution conjecture for gluing small Kerr black holes into initial data

Let $(\hat\gamma,\hat k)$ denote boosted subextremal Kerr black hole initial data with parameters $\hat{\mathfrak{m}}$ (mass) and $\hat{\mathfrak{a}}$ (specific angular momentum). Let $(X,\gamma,k)$ be initial data, and let $(M,g)$ be a compact globally hyperbolic subset of its maximal globally hyperbolic development. For a suitable family $(\gamma_\epsilon,k_\epsilon)$ of initial data with boundary data $(\gamma,k)$ and $(\hat\gamma,\hat k)$, write $(X_\epsilon,\gamma_\epsilon,k_\epsilon)$ for the glued initial data, and let $\mathcal C\subset M$ be the geodesic whose initial conditions at $X\subset M$ are determined by a point $\mathfrak p\in X$ and the boost parameter of the Kerr data. Evolution conjecture for gluing small Kerr black holes. The maximal globally hyperbolic development of $(X_\epsilon,\gamma_\epsilon,k_\epsilon)$ contains a region $(M_\epsilon,g_\epsilon)$ such that: (1) $M_\epsilon$ is obtained from $M$ by excising a size-$\epsilon$ neighborhood of $\mathcal C$; (2) $g_\epsilon$ tends to $g$ away from $\mathcal C$, while in an $\mathcal O(\epsilon)$ neighborhood of $\mathcal C$, the rescaling $\epsilon^{-2}g_\epsilon$ tends to a family, depending on the point of $\mathcal C$, of Kerr black hole metrics with parameters $(\hat{\mathfrak m},\hat{\mathfrak a})$; and (3) the total family $\epsilon\mapsto g_\epsilon$ is polyhomogeneous on a total space obtained by resolving $[0,1)\times M$ at $\{0\}\times\mathcal C$. This conjecture predicts that the local Kerr geometry inserted into the initial data persists under evolution, with the spacetime developing a controlled polyhomogeneous structure after the small-hole limit is resolved. Its resolution would describe how isolated boosted subextremal Kerr black holes emerge inside evolutions of glued initial data.