Equality in the logarithmic Sobolev inequality
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866914945024655360 |
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| author | Ohta, Shin-ichi Takatsu, Asuka |
| author_facet | Ohta, Shin-ichi Takatsu, Asuka |
| contents | We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_09400 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Equality in the logarithmic Sobolev inequality Ohta, Shin-ichi Takatsu, Asuka Differential Geometry Functional Analysis We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems. |
| title | Equality in the logarithmic Sobolev inequality |
| topic | Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/1904.09400 |