Unbounded normalized scalar curvature integrals in dimension four

Abstract

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 \[ \lim_{R\to\infty}R^{2-n}\int_{B_q(R)}\mathrm{Scal}_g\,\mathrm dV_g=+\infty \] for every fixed $q\in {\mathbb{R}}^n$. Here $B_q(R)$ denotes the geodesic ball of radius $R$ centered at $q$, and ${\mathrm{Scal}}_g$ is the scalar curvature of $g$.

20 page, 2 figures