Bakry-Èmery, Hardy, and Spectral Gap Estimates on Manifolds with Conical Singularities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929380776738816 |
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| author | Sturm, Karl-Theodor |
| author_facet | Sturm, Karl-Theodor |
| contents | We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $\ge3$ with (finite or countably many) conical singularities $\{z_i\}_{i\in\mathfrak I}$ in the neighborhood of which the largest lower bound for the Ricci curvature is \begin{equation}\label{d2} k(x)\simeq K_i-\frac{s_i}{d^2(z_i,x)}. \end{equation} Thus none of the existing Bakry-Émery inequalities or curvature-dimension conditions apply. In particular, $k$ does not belong to the Kato (or (extended Kato) class, and $(M,g)$ is not tamed. Manifolds with such a singular Ricci bound appear quite naturally., e.g. as cones over spheres of radius $>1$
For such manifolds with conical singularities we will prove
* a version of the Bakry-Émery inequality
* a novel Hardy inequality
* a spectral gap estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10734 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bakry-Èmery, Hardy, and Spectral Gap Estimates on Manifolds with Conical Singularities Sturm, Karl-Theodor Differential Geometry Metric Geometry We study spectral properties and geometric functional inequalities on Riemannian manifolds of dimension $\ge3$ with (finite or countably many) conical singularities $\{z_i\}_{i\in\mathfrak I}$ in the neighborhood of which the largest lower bound for the Ricci curvature is \begin{equation}\label{d2} k(x)\simeq K_i-\frac{s_i}{d^2(z_i,x)}. \end{equation} Thus none of the existing Bakry-Émery inequalities or curvature-dimension conditions apply. In particular, $k$ does not belong to the Kato (or (extended Kato) class, and $(M,g)$ is not tamed. Manifolds with such a singular Ricci bound appear quite naturally., e.g. as cones over spheres of radius $>1$ For such manifolds with conical singularities we will prove * a version of the Bakry-Émery inequality * a novel Hardy inequality * a spectral gap estimate. |
| title | Bakry-Èmery, Hardy, and Spectral Gap Estimates on Manifolds with Conical Singularities |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2405.10734 |