It is common that researchers study the critical metrics of Einstein-Hilbert like functional $${\cal R^2}=\int_M|{\rm Ric}_g|^2dV_g,\quad \text{or}\quad {\cal r^2}=\int_M{\rm scal}_g^2dV_g$$ and deduce some results. For example Michael Anderson in his paper
Anderson, Michael T., Extrema of curvature functionals on the space of metrics on 3-manifolds. II, Calc. Var. Partial Differ. Equ. 12, No. 1, 1-58 (2001). ZBL1018.53020.
has proved the following:
Theorem 0.1. Let $(N, g)$ be a complete $\cal R^2$ critical metric with non-negative scalar curvature. Then $(N,g)$ is flat.
I haven't read the full paper. My question is that how this theorem can be useful while it deals only with critical metrics? I mean how can one use this and similar results to deduce nice results that are true without critical metrics assumption?
I know that Einstein metrics are critical points of Einstein-Hilbert functional $r$ so I think one can replace critical metric with Einstein manifolds in similar theorems and the result will remain true. Is this the only possible application of these set of theorems that deals with critical metrics?