We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $$\mathrm {Ric}_{\infty } \ge K>0$$ Ric ∞ ≥ 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.

中文翻译：

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对数 Sobolev 不等式中的等式

我们研究了满足 $$\mathrm {Ric}_{\infty } \ge K>0$$ Ric ∞ ≥ K > 0 的加权黎曼流形上的对数 Sobolev 不等式的刚性问题。假设等式成立，我们证明一维高斯空间必然分裂，类似于Cheng-Zhou 在光谱间隙上的刚性结果以及Morgan 在等周不等式上的刚性结果。证明的关键要素是 Klartag 在黎曼流形上引入的针分解方法。我们还提出了几个相关的开放性问题。