On local rigidity theorems with respect to the scalar curvature
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914772691189760 |
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| author | Cheng, Liang |
| author_facet | Cheng, Liang |
| contents | By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an $n$-dimensional Riemannian manifold satisfies $$ \int_V R(g) dvol_g \ge 0 \text{\ \ and\ \ } I(V)\ge I(\mathbb{R}^n), $$ or $$ \int_V R(g) dvol_g \ge 0 \text{\ \ and\ \ } S(V)\ge S(\mathbb{R}^n),$$ then $g=g_{\mathbb{R}^n}$ on $V$, where $R(g)$ is the scalar curvature of $g$, $\mathbb{R}^n$ is Euclidean space, $ I(V)$ is the isoperimetric constant of $V$ and $S(V)$ is best constant of $L^2$ logarithmic Sobolev inequality of $V$. Moreover,we also obtain the local $\mathbb{R}^n$-rigidity about local Perelman's $ν$-entropy, and local $\mathbb{S}^n$-rigidity (resp. $\mathbb{H}^n$-rigidity) theorems regarding the cases concerning $R(g)\ge n(n-1)$ (resp. $R(g)\ge -n(n-1) $), weighted isoperimetric constant and best constant of weighted $L^2$ logarithmic Sobolev inequality for the weighted metric $\left(\cos{\frac{d_g(p,x)}{2}}\right)^{-4}g$ (resp. $\left(\cosh{\frac{d_g(p,x)}{2}}\right)^{-4}g$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_05011 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On local rigidity theorems with respect to the scalar curvature Cheng, Liang Differential Geometry Primary 53E20, Secondary 53C20 By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an $n$-dimensional Riemannian manifold satisfies $$ \int_V R(g) dvol_g \ge 0 \text{\ \ and\ \ } I(V)\ge I(\mathbb{R}^n), $$ or $$ \int_V R(g) dvol_g \ge 0 \text{\ \ and\ \ } S(V)\ge S(\mathbb{R}^n),$$ then $g=g_{\mathbb{R}^n}$ on $V$, where $R(g)$ is the scalar curvature of $g$, $\mathbb{R}^n$ is Euclidean space, $ I(V)$ is the isoperimetric constant of $V$ and $S(V)$ is best constant of $L^2$ logarithmic Sobolev inequality of $V$. Moreover,we also obtain the local $\mathbb{R}^n$-rigidity about local Perelman's $ν$-entropy, and local $\mathbb{S}^n$-rigidity (resp. $\mathbb{H}^n$-rigidity) theorems regarding the cases concerning $R(g)\ge n(n-1)$ (resp. $R(g)\ge -n(n-1) $), weighted isoperimetric constant and best constant of weighted $L^2$ logarithmic Sobolev inequality for the weighted metric $\left(\cos{\frac{d_g(p,x)}{2}}\right)^{-4}g$ (resp. $\left(\cosh{\frac{d_g(p,x)}{2}}\right)^{-4}g$). |
| title | On local rigidity theorems with respect to the scalar curvature |
| topic | Differential Geometry Primary 53E20, Secondary 53C20 |
| url | https://arxiv.org/abs/2310.05011 |