Let $(M^n,g)$ be an asymptotically flat manifold with order $\tau >0$. Specifically, there exists a compact set $K \subset M$ such that outside of $K$ there exists a $C^{\infty}$ diffeomorphism $\Phi:M \backslash K \to \mathbb{R}^n \backslash B(0,R)$ such that under this identification, \begin{equation} g_{ij} =\delta_{ij}+O(r^{-\tau}), \quad \partial^{|k|}g_{ij} =O(r^{-\tau-|k|}) \end{equation} for any partial derivative of order $k$ as $r \to \infty$, where $r$ is the Euclidean distance function.
How can we find a conformal transformation $e^{-f}g$ of $g$ such that it is a compact smooth Riemannian manifold when we add one point?