Equality in the logarithmic Sobolev inequality (Q2309429)

Let \((M,g)\) be a complete connected Riemannian manifold with a probability measure \(\omega\) which is a weighted multiple of the volume measure. Under suitable assumption of positivity of the weighted Ricci curvature \(\text{Ric}_\infty \geq K\), the purpose of the paper is to study the logarithmic Sobolev inequality. The authors show that, if the equality is attained by a non-negative locally Lipschitz function \(\rho\), then \((M,g,\omega)\) can be written as the product of the real line weighted with a \(K\)-Gaussian measure and another weighted manifold \((N,g_N,\omega_N)\). Additionally, \(\rho\) is a translation of \(\omega\) in the direction of the real line, that is, it is the density of the push-forward of \(\omega\) with respect to the translation by some \(t \in \mathbb{R}\) on the line. The main tool used in the proof of the theorem is given by the needle decomposition introduced by \textit{B. Klartag} [Needle decompositions in Riemannian geometry. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1457.53028)]. The decomposability into needles, together with the equality for the logarithmic Sobolev inequality, imply the same equality for needles. Hence, by restricting to needles, one shows that \(h=\log \rho\) is either constant or affine. In both cases, the equality is attained for Poincaré inequality and this allows to conclude.