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

Abstract

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:$$\operatorname{Sc}(x) \geq n(n-1)K \quad \text{for all } x \in V,$$ (b) The isoperimetric profile of $V$ is no less than that of space form $M^n_K$:$$ \operatorname{I}(V,\beta) := \inf_{\substack{\Omega\subset V \\ \mathrm{Vol}(\Omega)=\beta}} \mathrm{Area}(\partial \Omega) \geq \operatorname{I}(M^n_K,\beta) \quad \text{for some } \beta_0>0 \text{ and all } 0<\beta<\beta_0,$$\textbf{then} the sectional curvature of $V$ must satisfy $$\operatorname{Sec}(x) = K \quad \text{for all } x \in V.$$ Additionally, we derive some new scalar curvature rigidity theorems concerninglogarithmic Sobolev inequality and Perelman's $\boldsymbol{\mu}$-functional.