Scalar curvature growth on nonnegatively curved three-manifolds

Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, $$\underset{r\to \mathrm{\infty }}{\lim }\frac{1}{r}{\int }_{B_p(r)}\mathrm{Scal}dV=8\pi (\chi (M)-V_M)\le 8\pi (1-V_M),$$ where $V_M$ is the asymptotic volume ratio. In the one-ended case, we show that $\chi (M)\in \{0,1\}$. Thus positive asymptotic volume ratio gives the value $8\pi (1-V_M)$, while in the collapsed case the value is determined by the topology of the end. No pole or scalar-curvature bound is assumed. The proof combines separate smooth approximations of the Busemann and distance functions, integrable negative curvature errors, and a determinant estimate on the level surfaces. An averaged boundary estimate then gives convergence of the integrated extrinsic curvature. In the two-ended case, the splitting theorem gives the exact limit $8\pi \chi (N)$ for the compact surface factor $N$. The one-ended upper bound is attained for every prescribed asymptotic volume ratio in $[0,1]$, and the two-ended bound is also sharp.