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

In this paper, we obtain that logarithmic Sobolev and $\mathcal{W}$- functionals have fantastic power series expansion formulas when we choose suitable test functions. By using these power series expansion formulas, we prove that if for some open subset $V$ in an $n$-dimensional manifold satisfying $$ \frac{ \int_V R d\mu}{\mathrm{Vol}(V)} \ge n(n-1)K$$ and the isoperimetric profile of $V$ satisfying $$ \operatorname{I}(V,\beta)\doteq \inf\limits_{\Omega\subset V,\mathrm{Vol}(\Omega)=\beta}\mathrm{Area}(\partial \Omega) \ge \operatorname{I}(M^n_K,\beta),$$ for all $\beta<\beta_0$ and some $\beta_0>0$, where $R$ is the scalar curvature and $M^n_K$ is the space form of constant sectional curvature $K$,then $\operatorname{Sec}(x)=K$ for all $x\in V$. We also get several other new scalar curvature rigidity theorems regarding isoperimetric profile, logarithmic Sobolev inequality and Perelman's $\boldsymbol{\mu}$-functional.