Proposition: Finite-point invisibility

Let $F$ be a finite set in an open comparison region $U\subset {\mathbb{R}}^m$. There are smooth conformal metric families that agree with $\eta$ on neighborhoods of every point of $F$, have the same pointwise cones everywhere, and have nonzero first-order Einstein curvature elsewhere.

Proof. Choose $z\in U\setminus F$ and a ball about $z$ whose closure is contained in $U$ and misses $F$. Choose a smooth compactly supported bump $\chi$ in that ball, equal to one near $z$. Set

The metrics equal $\eta$ near $F$, and their cones are unchanged. Near $z$, Proposition prop:conformal gives the nonzero spatial components $-2(m-2)$ of their first variation's Einstein tensor.