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

TL;DR

This paper derives power series expansions for logarithmic Sobolev and $ ext{W}$-functionals, leading to new scalar curvature rigidity results and insights into geometric inequalities on manifolds.

Contribution

It introduces power series expansions for key functionals and applies them to establish scalar curvature rigidity theorems under isoperimetric and curvature bounds.

Findings

Power series expansions of logarithmic Sobolev and $ ext{W}$-functionals are obtained.

Scalar curvature rigidity is proved under isoperimetric and curvature conditions.

New results on scalar curvature rigidity related to logarithmic Sobolev inequality and Perelman's $oldsymbol{ u}$-functional.

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.