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Equality in the logarithmic Sobolev inequality

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Abstract

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying \(\mathrm {Ric}_{\infty } \ge K>0\). Assuming that 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.

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Acknowledgements

SO was supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K04844, 17H02846. AT was supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI) 15K17536, 16KT0132.

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Authors and Affiliations

  1. Department of Mathematics, Osaka University, Osaka, 560-0043, Japan

    Shin-ichi Ohta

  2. Department of Mathematical Sciences, Tokyo Metropolitan University, Tokyo, 192-0397, Japan

    Asuka Takatsu

  3. RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Tokyo, 103-0027, Japan

    Shin-ichi Ohta & Asuka Takatsu

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  1. Shin-ichi Ohta
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  2. Asuka Takatsu
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Correspondence to Shin-ichi Ohta.

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Ohta, Si., Takatsu, A. Equality in the logarithmic Sobolev inequality. manuscripta math. 162, 271–282 (2020). https://doi.org/10.1007/s00229-019-01134-9

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  • DOI: https://doi.org/10.1007/s00229-019-01134-9

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