Limit theorems for the total scalar curvature

We study some preservation phenomena for lower bound of total scalar curvatures on a smooth manifold. In particular, we prove that the lower bound of the weighted total scalar curvature (which is known as Perelman's $\mathcal{F}$-functional) on a closed $n$-manifold is preserved under the $W^{1, p}~(p > n^{2}/2)$-convergence of Riemannian metrics and uniformly $C^{0}$-convergence of potential functions, provided that each scalar curvature is nonnegative. In the proof, we used a certain stability of the Ricci flow and the heat flow with the Ricci flow background. We also give some examples that may provide clues to identify the weakest topology for such a preservation phenomenon of the lower bound.