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arXiv · 1904.09400

Equality in the logarithmic Sobolev inequality

Shin-ichi Ohta·Asuka Takatsu
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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 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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Shin-ichi Ohta, Asuka Takatsu. 2019-04-20. Equality in the logarithmic Sobolev inequality. https://arxiv.org/abs/1904.09400

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