Unbounded normalized scalar curvature integrals in dimension four

We give counterexamples to Yau’s question on normalized scalar curvature integrals in every dimension $n\ge 4$. For each such $n$, there exists a smooth complete Riemannian metric $g$ on ${\mathbb{R}}^n$ with nonnegative Ricci curvature and a pole such that $$\underset{R\to \mathrm{\infty }}{\lim }{R}^{2-n}{\int }_{B_q(R)}{\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}}_g\mathrm{d}{V}_g=+\mathrm{\infty }$$ for every fixed $q\in {\mathbb{R}}^n$. Here $B_q(R)$ denotes the geodesic ball of radius $R$ centered at $q$, and ${\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}}_g$ is the scalar curvature of $g$.