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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.06117 |
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Table of Contents:
- 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 $M^n_K$:$$ \operatorname{I}(V,β) := \inf_{\substack{Ω\subset V \\ \mathrm{Vol}(Ω)=β}} \mathrm{Area}(\partial Ω) \geq \operatorname{I}(M^n_K,β) \quad \text{for some } β_0>0 \text{ and all } 0<β<β_0,$$\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 $\boldsymbolμ$-functional.