The power series expansions of logarithmic Sobolev, $\mathcal{W}$- functionals and scalar curvature rigidity

In this paper, we obtain that the logarithmic Sobolev and $\mathcal{W}$-functionals admit remarkable power series expansions when appropriate test functions are selected. Using these expansions formulas, we prove that for an open subset $V$ in an $n$-dimensional manifold $M$ with $\bar{V}\subset M$ satisfying: (a)The scalar curvature of $V$ satisfies the lower bound:$$\mathrm{Sc}(x)\ge n(n-1)K\text{for all }x\in V,$$ (b) The isoperimetric profile of $V$ is no less than that of space form $M_K^n$:$$\mathrm{I}(V,\beta ):=\underset{\begin{array}{c}\mathrm{\Omega }\subset V\\ \mathrm{V}\mathrm{o}\mathrm{l}(\mathrm{\Omega })=\beta \end{array}}{\inf }\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(\mathrm{\partial }\mathrm{\Omega })\ge \mathrm{I}(M_K^n,\beta )\text{for some }{\beta }_0>0\text{ and all }0<\beta <{\beta }_0,$$$\mathbf{\text{then}}$ the sectional curvature of $V$ must satisfy $$\mathrm{Sec}(x)=K\text{for all }x\in V\mathrm{.}$$ Additionally, we derive some new scalar curvature rigidity theorems concerninglogarithmic Sobolev inequality and Perelman’s $\mathit{\mu }$-functional.