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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.06117 |
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| _version_ | 1866913847375298560 |
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| author | Cheng, Liang |
| author_facet | Cheng, Liang |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The power series expansions of logarithmic Sobolev, $\mathcal{W}$- functionals and scalar curvature rigidity Cheng, Liang Differential Geometry 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. |
| title | The power series expansions of logarithmic Sobolev, $\mathcal{W}$- functionals and scalar curvature rigidity |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2409.06117 |