  We study strictly positive solutions to the critical Laplace equation \[ -
\Delta u = n(n-2) u^{\frac{n+2}{n-2}}, \] decaying at most like $d(o,
x)^{-(n-2)/2}$, on complete noncompact manifolds $(M, g)$ with nonnegative
Ricci curvature, of dimension $n \geq 3$. We prove that, under an additional
mild assumption on the volume growth, such a solution does not exist, unless
$(M, g)$ is isometric to $\mathbb{R}^n$ and $u$ is a Talenti function. The
method employs an elementary analysis of a suitable function defined along the
level sets of $u$.

Analysis of PDEs;Differential Geometry